lagrangian and dirac constraints for the ideal ...morrison/20_mapiic_arx.pdfsitael s.p.a., pisa,...
TRANSCRIPT
Lagrangian and Dirac constraints for the ideal incompressible fluid and
magnetohydrodynamics
P. J. Morrison∗
Department of Physics and Institute for Fusion Studies,
The University of Texas at Austin, Austin, TX 78712-1060, USA
T. Andreussi
SITAEL S.p.A., Pisa, 56121, Italy
F. Pegoraro
Dipartimento di Fisica E. Fermi, Pisa, 56127, Italy
Abstract
The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian vari-
able description using his method of multipliers in the Lagrangian (variational) formulation. An alternative
is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac’s con-
straint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility
constraint using Dirac’s method in terms of both the canonical Poisson brackets in the Lagrangian variable
description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of
density. Both cases give dynamics of infinite-dimensional geodesic flow on the group of volume preserving
diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables
is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various
methods is made.
1
I. INTRODUCTION
A. Background
Sometimes constraints are maintained effortlessly, an example being ∇ · B = 0 in electrody-
namics which if initially true remains true, while alternatively in most cases dynamical equations
must be modified to maintain constraints, an example being ∇·v = 0 in fluid mechanics. The need
to apply constraints arises in a variety of contexts, ranging from gauge field theories [e.g. 83] to
optimization and control [e.g. 54]. A very common approach is to use the method of Lagrange
multipliers, which is taught in standard physics curricula for imposing holonomic constraints in
mechanical systems. Alternatively, Dirac [62], in pursuit of his goal of quantizing gauge field
theories, introduced a method that uses the Poisson bracket.
The purpose of the present article is to explore different methods for imposing the compress-
ibility constraint in ideal (dissipation-free) fluid mechanics and its extension to magnetohydrody-
namics (MHD), classical field theories intended for classical purposes. This endeavor is richer
than might be expected because the different methods of constraint can be applied to the fluid in
either the Lagrangian (material) description or the Eulerian (spatial) description, and the constraint
methods have different manifestations in the Lagrangian (action principle) and Hamiltonian for-
mulations. Although Lagrange’s multiplier is widely appreciated, it is not so well known that he
used it long ago for imposing the incompressibility constraint for a fluid in the Lagrangian vari-
able description [69]. More recently, Dirac’s method was applied for imposing incompressibility
within the Eulerian variable description, first in Nguyen & Turski [77, 78] and followed up in
several works [55, 56, 57, 75, 84]. Given that a Lagrangian conservation law is not equivalent to
an Eulerian conservation law, it remains to elucidate the interplay between the methods of con-
straint and the variables used for the description of the fluid. Thus we have three dichotomies: the
Lagrangian vs. Eulerian fluid descriptions, Lagrange multiplier vs. Dirac constraint methods, and
Lagrangian vs. Hamiltonian formalisms. It is the elucidation of the interplay between these, along
with generalizing previous results, that is the present goal.
It is well known that a free particle with holonomic constraints, imposed by the method of
Lagrange multipliers, is a geodesic flow. Indeed, Lagrange essentially observed this in Lagrange
[69] for the ideal fluid when he imposed the incompressibility constraint by his method. Lagrange
did this in the Lagrangian description by imposing the constraint that the map from initial positions
2
of fluid elements to their positions at time t preserves volume, and he did this by the method of
Lagrange multipliers. It is worth noting that Lagrange knew the Lagrange multiplier turns out to be
the pressure, but he had trouble solving for it. Lagrange’s calculation was placed in a geometrical
setting by Arnold [50] (see also Appendix 2 of [51]), where the constrained maps from the initial
conditions were first referred to as volume preserving diffeomorphisms in this context. Given this
background, in our investigation of the three dichotomies described above we emphasize geodesic
flow.
For later use we record here the incompressible Euler equations for the case with constant
density and the case where density is advected. The equations of motion, allowing for density
advection, are given by
∂v∂t
= −v · ∇v −1ρ∇p , (1)
∇ · v = 0 , (2)
∂ρ
∂t= −v · ∇ρ , (3)
where v(x, t) is the velocity field, ρ(x, t) is the mass density, p(x, t) is the pressure, and x ∈ D,
the region occupied by the fluid. These equations are generally subject to the free-slip boundary
condition n · v|∂D = 0, where n is normal to the boundary of D. The pressure field that enforces
the constraint (2) is obtained by setting ∂(∇ · v)∂t = 0, which implies
∆ρp := ∇ ·(1ρ∇p
)= −∇ · (v · ∇v) . (4)
For reasonable assumptions on ρ and boundary conditions, (4) is a well-posed elliptic equation
[see e.g. 63], so we can write
p = −∆−1ρ ∇ · (v · ∇v) . (5)
For the case where ρ is constant we have the usual Green’s function expression
p(x, t) = − ρ
∫d3x′G(x, x′)∇′ · (v′ · ∇′v′) , (6)
where G is consistent with Neumann boundary conditions [79] and v′ = v(x′, t). Insertion of (6)
into (1) gives∂v∂t
= −v · ∇v + ∇
∫d3x′G(x, x′)∇′ · (v′ · ∇′v′) , (7)
which is a closed system for v(x, t).
For MHD, equation (1) has the additional term (∇ × B) × B/ρ added to the righthand side.
Consequently for this model, the source of (5) is modified by the addition of this term to −v · ∇v.
3
B. Overview
Section II contains material that serves as a guide for navigating the more complicated calcu-
lations to follow. We first consider the various approaches to constraints in the finite-dimensional
context in Sections II A–II C. Section II A briefly covers conventional material about holonomic
constraints by Lagrange multipliers – here the reader is reminded how the free particle with holo-
nomic constraints amounts to geodesic flow. Section II B begins with the phase space action
principle, whence the Dirac bracket for constraints is obtained by Lagrange’s multiplier method,
but with phase space constraints as opposed to the usual holonomic configuration space constraints
used in conjunction with Hamilton’s principle of mechanics, as described in Section II A. Next, in
Section II C, we compare the results of Sections II A and II C and show how conventional holo-
nomic constraints can be enforced by Dirac’s method. Contrary to Lagrange’s method, here we
obtain explicit expressions, ones that do not appear in conventional treatments, for the Christoffel
symbol and the normal force entirely in terms of the original Euclidean coordinates and con-
straints. Section II is completed with Section II D, where the previous ideas are revisited in the
d+1 field theory context in prepartion for the fluid and MHD calculations. Holonomic constraints,
Dirac brackets, with local or nonlocal constraints, and geodesic flow are treated.
In Section III we first consider the compressible (unconstrained) fluid and MHD versions of
Hamilton’s variational principle, the principle of least action, with Lagrange’s Lagrangian in the
Lagrangian description. From this we obtain in Section III B the canonical Hamiltonian field
theoretic form in the Lagrangian variable description, which is transformed in Section III C, via
the mapping from Lagrangian to Eulerian variables, to the noncanonical Eulerian form. Section
III is completed by an in depth comparison of constants of motion in the Eulerian and Lagrangian
descriptions, which surprisingly does not seem to appear in fluid mechanics or plasma physics
textbooks. The material of this section is necessary for understanding the different manifestations
of constraints in our dichotomies.
Section IV begins with Section IV A that reviews Lagrange’s original calculations. Because
the incompressibility constraint he imposes is holonomic and there are no additional forces, his
equations describe infinite-dimensional geodesics flow on volume preserving maps. The remain-
ing portion of this section contains the most substantial calculations of the paper. In Section IV B
for the first time Dirac’s theory is applied to enforce incompressibility in the Lagrangian variable
description. This results in a new Dirac bracket that generates volume preserving flows. As in Sec-
4
tion II C 1, which serves as a guide, the equations of motion generated by the bracket are explicit
and contain only the constraints and original variables. Next, in Section IV C, a reduction from
Lagrangian to Eulerian variables is made, resulting in a new Eulerian variable Poisson bracket that
allows for density advection while preserving incompressibility. This was an heretofore unsolved
problem. Section IV D ties together the results of Sections IV B, IV C, and III D. Here both the
Eulerian and Lagrangian Dirac constraint theories are compared after they are evaluated on their
respective constraints, simplifying their equations of motion. Because Lagrangian and Eulerian
conservation laws are not identical, we see that there are differences. Section IV concludes in
Section IV E with a discussion of the full algebra of invariants, that of the ten parameter Galilean
group, for both the Lagrangian and Eulerian descriptions. In addition the Casimir invariants of the
theories are discussed.
The paper concludes with Section V, where we briefly summarize our results and speculate
about future possibilities.
II. CONSTRAINT METHODS
A. Holonomic constraints by Lagrange’s multiplier method
Of interest are systems with Lagrangians of the form L(q, q) where the overdot denotes time
differentiation and q = (q1, q2, . . . , qN). Because nonautonomous systems could be included by
appending an additional degree of freedom, explicit time dependence is not included in L.
Given the Lagrangian, the equations of motion are obtained according to Hamilton’s principle
by variation of the action
S [q] =
∫ t1
t0dt L(q, q) ; (8)
i.e.
δS [q; η] :=ddε
S [q + εδq]∣∣∣∣ε=0
=
∫ t1
t0dt
(ddt∂L∂qi −
∂L∂qi
)δqi =
∫ t1
t0dtδS [q]δqi(t)
δqi = 0 , (9)
for all variations δq(t) satisfying δq(t0) = δq(t1) = 0, implies Lagrange’s equations of motion, i.e.
δS [q]δqi(t)
= 0 ⇒ddt∂L∂qi −
∂L∂qi = 0 , i = 1, 2, . . . ,N . (10)
Holonomic constraints are real-valued functions of the form CA(q), A = 1, 2, . . . ,M, which are
desired to be constant on trajectories. Lagrange’s method for implementing such constraints is to
5
add them to the action and vary as follows:
δS λ := δ(S + λACA) = 0 , (11)
yielding the equations of motion
δS λ[q]δqi(t)
= 0 ⇒ddt∂L∂qi −
∂L∂qi = λA
∂CA
∂qi , i = 1, 2, . . . ,N , (12)
with the forces of constraint residing on the right-hand side of (12). Observe in (11) and (12)
repeated sum notation is implied for the index A. The N equations of (12) with the M numerical
values of the constraints CA(q) = CA0 , determine the N + M unknowns {qi} and {λA}. In practice,
because solving for the Lagrange multipliers can be difficult an alternative procedure, a example
of which we describe in Sec. II A 1, is used.
We will see in Section IV A that the field theoretic version of this method is how Lagrange
implemented the incompressiblity constraint for fluid flow. For the purpose of illustration and in
preparation for later development, we consider a finite-dimensional analog of Lagrange’s treat-
ment.
1. Holonomic constraints and geodesic flow via Lagrange
Consider N noninteracting bodies each of mass mi in the Eucledian configuration space E3N
with cartesian coordinates qi = (qxi, qyi, qzi), where i = 1, 2, . . . ,N. The Lagrangian for this system
is given by the usual kinetic energy,
L =
N∑i=1
mi
2qi · qi , (13)
with the usual “dot” product. The Euler-Lagrange equations for this system, equations (10), are
the uninteresting system of N free particles. As in Sec. II A we constrain this system by adding
constraints CA(q1,q2, . . . ,qN), where again A = 1, 2, . . . ,M, leading to the equations
miqi = λA∂CA
∂qi. (14)
Instead of solving the 3N equations of (14) together with the M numerical values of the con-
straints, in order to determine the unknowns qi and λA, we recall the alternative procedure, which
dates back to Lagrange (see e.g. Sec. IV of Lagrange [69]) and has been taught to physics students
6
for generations [see e.g. 59, 86]. With the alternative procedure one introduces generalized coor-
dinates that account for the constraints, yielding a smaller system on the constraint manifold, one
with the Lagrangian
L =12
gµν(q) qµqν , µ, ν = 1, 2 . . . , 3N − M , (15)
where
gµν =
N∑i=1
mi∂qi
∂qµ·∂qi
∂qν. (16)
Then Lagrange’s equations (10) for the Lagrangian (15) are the usual equations for geodesic flow
qµ + Γµαβ qαqβ = 0 , (17)
where the Christoffel symbol is as usual
Γµαβ =
12
gµν(
gνα∂qβ
+gνβ∂qα−
gαβ∂qν
). (18)
If the constraints had time dependence, then the procedure would have produced the Coriolis and
centripetal forces, as is usually done in textbooks.
Thus, we arrive at the conclusion that free particle dynamics with time-independent holonomic
constraints is geodesic flow.
B. Dirac’s bracket method
So, a natural question to ask is “How does one implement constraints in the Hamiltonian set-
ting, where phase space constraints depend on both the configuration space coordinate q and the
canonical momentum p”? [See e.g. 53, 83]. To this end we begin with the phase space action
principle
S [q, p] =
∫ t1
t0dt
[piqi − H(q, p)
], (19)
where again repeated sum notation is used for i = 1, 2, . . . ,N. Independent variation of S [q, p]
with respect to q and p, with δq(t0) = δq(t1) = 0 and no conditions on δp, yields Hamilton’s
equations,
pi = −∂H∂qi and qi =
∂H∂pi
, (20)
or equivalently
zα = {zα,H} , (21)
7
which is a rewrite of (20) in terms of the Poisson bracket on phase space functions f and g,
{ f , g} =∂ f∂qi
∂g∂pi−∂g∂qi
∂ f∂pi
=∂ f∂zαJαβc
∂g∂zβ
, (22)
where in the second equality we have used z = (q, p), so α, β = 1, 2, . . . , 2N and the cosymplectic
form (Poisson matrix) is
Jc =
ON IN
−IN ON
, (23)
with ON being an N × N block of zeros and IN being the N × N identity.
Proceeding as in Section II A, albeit with phase space constraints Da(q, p), a = 1, 2, . . . , 2M <
2N, we vary
S λ[q, p] =
∫ t1
t0dt
[piqi − H(q, p) + λaDa
], (24)
and obtain
pi = −∂H∂qi + λa
∂Da
∂qi and qi =∂H∂pi− λa
∂Da
∂pi. (25)
Next, enforcing Da = 0 for all a, will ensure that the constraints stay put. Whence, differentiating
the Da and using (25) yields
Da =∂Da
∂qi qi +∂Da
∂pipi
= {Da,H} − λb{Da,Db} ≡ 0 . (26)
We assume Dab := {Da,Db} has an inverse, D−1ab , which requires there be an even number of
constraints, a, b = 1, 2, . . . , 2M, because odd antisymmetric matrices have zero determinant. Then
upon solving (26) for λb and inserting the result into (25) gives
zα = {zα,H} − D−1ab {z
α,Da}{Db,H} . (27)
From (27), we obtain a generalization of the Poisson bracket, the Dirac bracket,
{ f , g}∗ = { f , g} − D−1ab { f ,D
a}{Db, g} . (28)
which has the degeneracy property
{ f ,Da}∗ ≡ 0 . (29)
for all functions f and indices a = 1, 2, . . . , 2M.
The generation of the equations of motion via a Dirac bracket, i.e.
zα = {zα,H}∗ , (30)
8
which is equivalent to (27), has the advantage that the Lagrange multipliers λA have been elimi-
nated from the theory.
Note, although the above construction of the Dirac bracket started with the canonical bracket
of (III C), his construction results in a valid Poisson bracket if one starts from any valid Poisson
bracket (cf. (83) of Section II D and Section III C), which need not have a Poisson matrix of the
form of (23) [see e.g. 75]. We will use such a bracket in Section IV C when we apply constraints
by Dirac’s method in the Eulerian variable picture. Also note, for our purposes it is not necessary
to describe primary vs. secondary constraints (although we use the latter), and the notions of weak
vs. strong equality. We refer the reader to Dirac [62], Hanson et al. [66], Sudarshan & Makunda
[82], Sundermeyer [83] for treatment of these concepts.
C. Holonomic constraints by Dirac’s bracket method
A connection between the approaches of Lagrange and Dirac can be made. From a set of La-
grangian constraints CA(q), where A = 1, 2, . . . ,M, one can construct an additional M constraints
by differentiation,
CA =∂CA
∂qi qi =∂CA
∂qi
∂H∂pi
, (31)
where the second equality is possible if (10) possesses the Legendre transformation to the Hamil-
tonian form. In this way we obtain an even number of constraints
Da(q, p) =(CA(q) , CA′(q, p)
), (32)
where A = 1, 2, . . . ,M, A′ = M + 1,M + 2, . . . , 2M, and a = 1, 2, . . . , 2M.
With the constraints of (32) the bracket Dab = {Da,Db} needed to construct (28) is easily
obtained,
D =
OM {CA, CB′}
{CA′ ,CB} {CA′ , CB′}
=:
OM S
−S A
, (33)
where OM is an M × M block of zeros and S is the following M × M symmetric matrix with
elements
SAB := {CA, CB} =∂2H∂pi∂p j
∂CA
∂qi
∂CB
∂q j , (34)
9
and A is the following M × M antisymmetric matrix with elements
AAB := {CA′ , CB′} (35)
=∂2H∂pi∂pk
[∂2H∂qi∂p j
(∂CA
∂q j
∂CB
∂qk −∂CB
∂q j
∂CA
∂qk
)+∂H∂p j
(∂2CA
∂qi∂q j
∂CB
∂qk −∂2CB
∂qi∂q j
∂CA
∂qk
)].
Assuming the existence of D−1, the 2M × 2M inverse of (33), the Dirac bracket of (28) can be
constructed. A necessary and sufficient condition for the existence of this inverse is that detS , 0,
and when this is the case the inverse is given by
D−1 =
S−1AS−1 −S−1
S−1 OM
. (36)
Because of the block diagonal structure of (36), the Dirac bracket (28) becomes
{ f , g}∗ = { f , g} + S−1AB
({ f ,CA}{CB, g} − {g,CA}{CB, f }
)+ S−1
AC ACD S−1
DB { f ,CA}{CB, g} , (37)
which has the form
{ f , g}∗ = { f , g} − (P⊥)ij
(∂ f∂qi
∂g∂p j−∂g∂qi
∂ f∂p j
)+ Qi j ∂ f
∂pi
∂g∂p j
=∂ f∂qi P
ij∂g∂p j−∂g∂qi P
ij∂ f∂p j
+ Qi j ∂ f∂pi
∂g∂p j
, (38)
where the matrices P = I − P⊥, with
(P⊥)ij = S−1
AB∂2H∂pi∂pk
∂CA
∂q j
∂CB
∂qk , (39)
and Q, a complicated expression that we will not record, are crafted using the constraints and
Hamiltonian so as to make { f , g}∗ preserve the constraints.
The equations of motion that follow from (37) are
q` = {q`,H}∗ =∂H∂p`
+ S−1AB
({q`,CA}{CB,H} − {H,CA}{CB, q`}
)+ S−1
AC ACD S−1
DB {q`,CA}{CB,H} , (40)
p` = {p`,H}∗ = −∂H∂q`
+ S−1AB
({p`,CA}{CB,H} − {H,CA}{CB, p`}
)+ S−1
AC ACD S−1
DB {p`,CA}{CB,H} . (41)
10
Given the Dirac bracket associated with the D of (34), dynamics that enforces the constraints
takes the form of (30). Any system generated by this bracket will enforce Lagrange’s holonomic
constraints; however, only initial conditions compatible with
Da ≡ 0 , ∀ a = M + 1,M + 2, . . . , 2M , (42)
or equivalently
CA =∂CA
∂qi
∂H∂pi
= {CA,H} ≡ 0 , ∀ A = 1, 2, . . . ,M , (43)
will correspond to the system with holonomic constraints. Using (43) and {q`,CA} ≡ 0, (40) and
(41) reduce to
q` = {q`,H}∗ =∂H∂p`
, (44)
p` = {p`,H}∗ = −∂H∂q`
+ S−1AB{p`,C
A}{CB,H} , (45)
where
{CB,H} =
(∂2H∂qi∂p j
∂CB
∂q j +∂H∂p j
∂2CB
∂qiq j
)∂H∂pi−∂CB
∂qi
∂2H∂pi∂p j
∂H∂q j . (46)
Thus the Dirac bracket approach gives a relatively simple system for enforcing holonomic con-
straints. It can be shown directly that if initially CA vanishes, then the system of (44) and (45) will
keep it so for all time.
1. Holonomic constraints and geodesic flow via Dirac
Let us now consider again the geodesic flow problem of Sec. II A 1: the N degree-of-freedom
free particle system with holonomic constraints, but this time within the framework of Dirac
bracket theory. For this problem the unconstrained configuration space is the Euclidean space
E3N and we will denote by Q the constraint manifold within E3N defined by the constancy of the
constraints CA.
The Lagrangian of (13) is easily Legendre transformed to the free particle Hamiltonian
H =
N∑i=1
12mi
pi · pi , (47)
where pi = miqi. For this example the constraints of (32) take the form
Da =
(CA(q1,q2, . . . ,qN),
∂CA′(q1,q2, . . . ,qN)∂qi
·pi
mi
), (48)
11
the M × M matrix S has elements
SAB =
N∑i=1
1mi
∂CA
∂qi·∂CB
∂qi, (49)
and the M × M matrix A is
AAB =
N∑i, j=1
1mim j
pi ·
[∂2CA
∂qi∂q j·∂CB
∂q j−
∂2CB
∂qi∂q j·∂CA
∂q j
]. (50)
The Dirac bracket analogous to (38) is
{ f , g}∗ =
N∑i j=1
[∂ f∂qi·↔
Pi j ·∂g∂p j−∂g∂qi·↔
Pi j ·∂ f∂p j
+∂ f∂pi·↔
Qi j ·∂g∂p j
], (51)
where↔
Pi j =↔
I i j −↔
P⊥ i j with the tensors
↔
P⊥ i j :=N∑
k=1
S−1AB
∂2H∂pi∂pk
·∂CB
∂qk
∂CA
∂q j= S−1
AB1mi
∂CB
∂qi
∂CA
∂q j, (52)
↔
Q i j :=N∑
k=1
S−1AB
[∂CA
∂q j
pk
mk·∂2CB
∂qk∂qi−∂CA
∂qi
pk
mk·∂2CB
∂qk∂q j
]+ S−1
AC ACD S−1
DB∂CA
∂qi
∂CB
∂q j(53)
=:↔
Ti j −↔
T ji +↔
Ai j , (54)
where↔
Ai j is the term with S−1ACA
CDS−1DB. Observe
↔
Ai j = −↔
A ji because ACD = −ADC and
N∑k=1
↔
P⊥ ik ·↔
P⊥ k j =
N∑k=1
(S−1
AB1mi
∂CB
∂qi
∂CA
∂qk
)·
(S−1
A′B′1
mk
∂CB′
∂qk
∂CA′
∂q j
)
=
(S−1
AB1mi
∂CB
∂qi
)S−1
A′B′
N∑k=1
∂CA
∂qk·
1mk
∂CB′
∂qk
∂CA′
∂q j
=
(S−1
AB1mi
∂CB
∂qi
)S−1
A′B′ SAB′ ∂CA′
∂q j=↔
P⊥ i j . (55)
Also observe for the Hamiltonian of (47)
N∑j=1
↔
P⊥ i j ·∂H∂p j
=
N∑j=1
↔
P⊥ i j ·p j
m j≡ 0 , (56)
N∑j=1
∂H∂p j·↔
Ti j =
N∑j=1
p j
m j·↔
Ti j ≡ 0 , (57)
N∑j=1
↔
Ai j ·∂H∂p j
= −
N∑j=1
↔
A ji ·∂H∂p j≡ 0 , (58)
12
when evaluated on the constraint CA,B = 0, while
N∑j=1
∂H∂p j·↔
P⊥ ji , 0 andN∑
i=1
∂H∂pi·↔
Ti j , 0 , (59)
when evaluated on the constraint CA,B = 0. Thus, the bracket of (51) yields the equations of motion
qi = {qi,H}∗ =∂H∂pi
=pi
mi, (60)
pi = −∂CA
∂qiS−1
AB
N∑j,k=1
p j
m j·∂2CB
∂q jqk·
pk
mk, (61)
or
qi = −1mi
∂CA
∂qiS−1
AB
N∑j,k=1
q j ·∂2CB
∂q jqk· qk = −
N∑j,k=1
q j · Γi, jk · qk , (62)
where
Γi, jk :=1mi
∂CA
∂qi⊗ S−1
AB∂2CB
∂q jqk, (63)
is used to represent the normal force.
Observe, (62) has two essential features: as noted, its righthand side is a normal force that
projects to the constraint manifold defined by the constraints CA and within the constraint manifold
it describes a geodesic flow, all done in terms of the original Euclidean space coordinates where
the initial conditions place the flow on Q by setting the values CA for all A = 1, 2, . . . ,M. We will
show this explicitly.
First, because the components of vectors normal toQ are given by ∂CA/∂qi for A = 1, 2, . . . ,M,
this prefactor on the righthand side of (62) projects as expected. Upon comparing (62) with (14)
we conclude that the coefficient of this prefactor must be the Lagrange multipliers, i.e.,
λA = −S−1AB
N∑k, j=1
q j ·∂2CB
∂q jqk· qk . (64)
Thus, we see that Dirac’s procedure explicitly solves for the Lagrange multiplier.
Second, to uncover the geodesic flow we can proceed as usual by projecting explicitly onto
Q. To this end we consider the transformation between the Euclidean configuration space E3N
coordinates
(q1,q2, . . . ,qi, . . . ,qN) , where i = 1, 2, . . . ,N (65)
and another set of coordinates
(q1, q2, . . . , qa, . . . , q3N) , where a = 1, 2, . . . , 3N , (66)
13
which we tailor as follows:
(q1, q2, . . . , qα . . . , qn,C1,C2, . . . ,CA, . . . ,CM) , (67)
where α = 1, 2, . . . , n, A = 1, 2, . . . ,M, and n = 3N − M. Here we have chosen qn+A = CA and n
is the actual number of degrees of freedom on the constraint surface Q. We can freely transform
back and forth between the two coordinates, i.e.
(q1,q2, . . . ,qi, . . . ,qN)←→ (q1, q2, . . . , qa, . . . , q3N) . (68)
Note, the choice qn+A = CA could be replaced by qn+A = f A(C1,C2, . . . ,CM) for arbitrary indepen-
dent f A, but we assume the original CA are optimal. Because qα are coordinates within Q, tangent
vectors to Q have the components ∂qi/∂qα, and there is one for each α = 1, 2, . . . , n. The pairing
of the normals with tangents is expressed by
N∑i=1
∂qi
∂qα·∂CA
∂qi= 0 , α = 1, 2, . . . , n; A = 1, 2, . . . ,M . (69)
Let us now consider an alternative procedure that the Dirac constraint method provides. Proceed-
ing directly we calculate
qa =
N∑i=1
∂qa
∂qi· qi . (70)
Observe that on E3N the matrix ∂qa/∂qi is invertible and the full metric tensor and its inverse in
the new coordinates are given as follows:
gab =
N∑i=1
1mi
∂qa
∂qi·∂qb
∂qiand gab =
N∑i=1
mi∂qi
∂qa ·∂qi
∂qb . (71)
The metric tensor on Q of (16) is obtained by restricting gab to a, b ≤ n and gαβ is obtained by
inverting gαβ and not by restricting gab. Proceeding by differentiating again we obtain
qa =
N∑i=1
∂qa
∂qi· qi +
N∑i, j=1
qi ·∂2qa
∂qi∂q j· q j , a = 1, 2, . . . , 3N. (72)
Now inserting (62) into (72) gives
qa = −
N∑i=1
1mi
∂qa
∂qi·∂CA
∂qigAB
N∑j,k=1
q j ·∂2CB
∂q jqk· qk +
N∑i, j=1
q j ·∂2qa
∂qi∂q j· qi , (73)
where we have recognized that
gAB = SAB =
N∑i=1
1mi
∂CA
∂qi·∂CB
∂qi(74)
14
and, as was necessary for the workability of the Dirac bracket constraint theory, gAB = S−1AB must
exist. This quantity is obtained by inverting SAB and not by restricting gab.
Equation (73) is an expression for the full system on E3N . However, for a > n, we know
qa = CA = 0, so the two terms of (73) should cancel. To see this, in the first term of (73) we set
qa = CC and observe that this first term becomes
−
N∑i=1
1mi
∂CC
∂qi·∂CA
∂qigAB
N∑j,k=1
q j ·∂2CB
∂q jqk· qk = −gCA gAB
N∑j,k=1
q j ·∂2CB
∂q jqk· qk
= −
N∑j,k=1
q j ·∂2CC
∂q jqk· qk . (75)
Now, for a ≤ n, say α, the righthand side gives a Christoffel symbol expression for the geodesic
flow; viz.
qα = −
N∑i=1
1mi
∂qα
∂qi·∂CA
∂qigAB
N∑j,k=1
q j ·∂2CB
∂q jqk· qk +
N∑j,k=1
q j ·∂2qα
∂q j∂qk· qk
= −Γαµν qµqν , (76)
where
Γαµν =
N∑i=1
1mi
∂qα
∂qi·∂CA
∂qiS−1
AB
N∑j,k=1
∂q j
∂qµ·∂2CB
∂q jqk·∂qk
∂qν+
N∑j,k=1
∂q j
∂qµ·∂2qα
∂q j∂qk·∂qk
∂qν(77)
is an expression for the Christoffel symbol in terms of the original Euclidean coordinates, the
constraints, and the choice of coordinates on Q.
Using (77) one can calculate an analogous expression for the Riemann curvature tensor on Q
from the usual expression
Rαβγδ =
∂Γαδβ
∂qγ−∂Γαγβ
∂qδ+ ΓαγλΓ
λγβ − ΓαδλΓ
λγβ , (78)
using ∂/∂qγ =∑
i(∂qi/∂qγ) · ∂/∂qi. This gives the curvature written in terms of the original
Euclidean coordinates, the constraints, and the chosen coordinates on Q.
D. d + 1 field theory
The techniques of Sections II A, II B, and II C have natural extensions to field theory.
Given independent field variables ΨA(µ, t), indexed byA = 1, 2, . . . , `, where the independent
variable µ = (µ1, µ2, . . . , µd). The field theoretic version of Hamilton’s principle of (8) is embodied
15
in the action
S [Ψ] =
∫ t1
t0dt L[Ψ, Ψ] , with L[Ψ, Ψ] =
∫ddµL(Ψ, Ψ, ∂Ψ) , (79)
where we leave the domain of µ and the boundary conditions unspecified, but freely drop surface
terms obtained upon integration by parts. The Lagrangian density L is assumed to depend on the
field components Ψ and ∂Ψ, which is used to indicate all possible partial derivatives with respect
of the components of µ. Hamilton’s principle with (79) gives the Euler-Lagrange equations,
δS [Ψ]δΨA(µ, t)
= 0 ⇒ddt
∂L∂ΨA
+∂
∂µ
∂L∂∂ΨA
−∂L∂ΨA
= 0 , (80)
where the overdot implies differentiation at constant µ. Local holonomic constraints CA(Ψ, ∂Ψ)
are enforced by Lagrange’s method by amending the Lagrangian
Lλ[Ψ, Ψ] =
∫ddµ
(L(Ψ, Ψ, ∂Ψ) + λACA(Ψ, ∂Ψ)
), (81)
with again A = 1, 2, . . . ,M and proceeding as in the finite-dimensional case.
In the Hamiltonian field theoretic setting, we could introduce the conjugate momentum densi-
ties πA,A = 1, 2, . . . , `, with the phase space action
Sλ[Ψ, π] =
∫ t1
t0dt∫
ddµ[πAΨA −H + λADA
], (82)
with Hamiltonian density H and local constraints Da depending on the values of the fields and
their conjugates. Instead of following this route we will jump directly to a generalization of the
field theoretic Dirac bracket formalism that would result.
Consider a Poisson algebra composed of functionals of field variables χA(µ, t) with a Poisson
bracket of the form
{F,G} =
∫ddµ Fχ · J(χ) ·Gχ , (83)
where Fχ is a shorthand for the functional derivative of a functional F with respect to the field χ
[see e.g. 72] and Fχ · J ·Gχ = FχA JABGχB , again with repeated indices summed. Observe the fields
χA(µ, t) need not separate into coordinates and momenta, but if they do the Poisson operator J has
a form akin to that of (23). By a Poisson algebra we mean a Lie algebra realization on functionals,
meaning the Poisson bracket is bilinear, antisymmetric, and satisfies the Jacobi identity and that
there is an associative product of functionals that satisfies the Leibniz law. From the Poisson
bracket the equations of motion are given by χ = {χ,H}, for some Hamiltonian functional H[χ].
16
Dirac’s constraint theory is generally implemented in terms of canonical Poisson brackets [see
e.g. 62, 82, 83], but it is not difficult to show that his procedure also works for noncanonical
Poisson brackets [see e.g. an Appendix of 75].
We impose an even number of local constraints which we write as Da(µ) = constant, a shorthand
for Da[χ(µ)], with the index a = 1, 2, . . . , 2M, bearing in mind that they depend on the fields χ and
their derivatives. As in the finite-dimensional case, the Dirac bracket is obtained from the matrix
D obtained from the bracket of the constraints,
Dab(µ, µ′) = {Da(µ),Db(µ′)} ,
where we note that Dab(µ, µ′) = −Dba(µ′, µ). If D has an inverse, then the Dirac bracket is defined
as follows:
{F,G}∗ = {F,G} −∫
ddµ
∫ddµ′ {F,Da(µ)}D−1
ab (µ, µ′){Db(µ′),G} , (84)
where the coefficients D−1ab (µ, µ′) satisfy∫
ddµ′D−1ab (µ, µ′)Dbc(µ′, µ′′) =
∫d3µ′Dcb(µ, µ′)D−1
ba (µ′, µ′′) = δcaδ(µ − µ
′′) ,
consistent with D−1ab (µ, µ′) = −D−1
ba (µ′, µ).
We note, the procedure is effective only when the coefficients D−1ab (µ, µ′) can be found. If D is
not invertible, then one needs, in general, secondary constraints to determine the Dirac bracket.
1. Field theoretic geodesic flow
In light of Sec. II A 1, any field theory with a Lagrangian density of the form
L =12
ΨA(µ, t) ηAB ΨB(µ, t) , (85)
with the metric ηAB = δAB being the Kronecker delta, subject to time-independent holonomic
constraints can be viewed as geodesic flow on the constraint surface. This is a natural infinite-
dimensional generalization of the idea of Section II A 1.
III. UNCONSTRAINED HAMILTONIAN AND ACTION FOR FLUID
A. Fluid action in Lagrangian variable description
The Lagrangian variable description of a fluid is described in Lagrange’s famous work [69],
while historic and additional material can be found in Morrison [72], Newcomb [76], Serrin [80],
17
Van Kampen & Felderhof [85]. Because the Lagrangian description treats a fluid as a continuum of
particles, it naturally is amenable to the Hamiltonian form. The Lagrangian variable is a coordinate
that gives the position of a fluid element or parcel, as it is sometimes called, at time t. We denote
this variable by q = q(a, t) = (q1, q2, q3), which is measured relative to some cartesian coordinate
system. Here a = (a1, a2, a3) denotes the fluid element label, which is often defined to be the
position of the fluid element at the initial time, a = q(a, 0), but this need not always be the case.
The label a is a continuum analog of the discrete index that labels a generalized coordinate in
a finite degree-of-freedom system. If D is a domain that is fully occupied by the fluid, then at
each fixed time t, q : D → D is assumed to be 1-1 and onto. Not much is really known about
the mathematical properties of this function, but we will assume that it is as smooth as it needs to
be for the operations performed. Also, we will assume we can freely integrate by parts dropping
surface terms and drop reference to D in our integrals.
When discussing the ideal fluid and MHD we will use repeated sum notation with upper and
lower indices even though we are working in cartesian coordinates. And, unlike in Section 2, latin
indices, i, j, k, ` etc. will be summed over 1,2, and 3, the cartesian components, rather than to N.
This is done to avoid further proliferation of indices and we trust confusion will not arise because
of context.
Important quantities are the deformation matrix, ∂qi/∂a j and its Jacobian determinant J :=
det(∂qi/∂a j), which is given by
J =16εk j`ε
imn∂qk
∂ai
∂q j
∂am
∂q`
∂an , (86)
where εi jk = ε i jk is the purely antisymmetric (Levi-Civita) tensor density. We assume a fluid
element is uniquely determined by its label for all time. Thus, J , 0 and we can invert q = q(a, t)
to obtain the label associated with the fluid element at position x at time t, a = q−1(x, t). For
coordinate transformations q = q(a, t) we have
∂qk
∂a j
Aik
J= δi
j , (87)
where Aik is the cofactor of ∂qk/∂ai , which can be written as follows:
Aik =
12εk j`ε
imn ∂q j
∂am
∂q`
∂an . (88)
Using q(a, t) or its inverse q−1(x, t), various quantities can be written either as a function of x
18
or a. For convenience we list additional formulas below for latter use:
J =13
Ak`
∂q`
∂ak , (89)
A ji =
∂J
∂(∂qi/∂a j), (90)
∂(Aki f )
∂ak = Aki∂ f∂ak , (91)
δJ = Aki∂δqi
∂ak or J = Aki∂qi
∂ak , (92)
δ
(Ak`
J
)∂q`
∂au = −Ak
i
J
∂
∂au δqi or δ
(Ak`
J
)= −
Aki Au
`
J2
∂
∂au δqi , (93)
Au`
∂
∂au
[Ak
i
J
∂z`
∂ak
]= Ak
i∂
∂ak
[Au`
J
∂z`
∂au
], ∀ z` , (94)
which follow from the standard rule for differentiation of determinants and the expression for the
cofactor matrix.
Now we are in a position to recreate and generalize Lagrange’s Lagrangian for the ideal fluid
action principle. On physical grounds we expect our fluid to possess kinetic and internal energies,
and if magnetized, a magnetic energy. The total kinetic energy functional of the fluid is naturally
given by
T [q] :=12
∫d3a ρ0(a) |q|2 , (95)
where ρ0 is the mass density attached to the fluid element labeled by a and q denotes time differ-
entiation of q at fixed label a. Note, in (95) |q|2 = qiqi, where in general qi = gi j qi, but we will
only consider the cartesian metric where gi j = δi j = ηi j.
Fluids are assumed to be in local thermodynamic equilibrium and thus can be described by a
function U(ρ, s), an internal energy per unit mass that depends on the specific volume ρ−1 and
specific entropy s. If a magnetic field B(x, t) were present, then we could add dependence on |B|
as in Morrison [71] to account for pressure anisotropy. The internal energy is written in terms
of the Eulerian density and entropy (see Section III C) since we expect the fluid at each Eulerian
observation point to be in thermal equilibrium. From U we compute the temperature and pressure
according to the usual differentiations, T = ∂U/∂s and p = ρ2∂U/∂ρ. For MHD, the magnetic
energy HB =∫
d3x |B|2/2 in Lagrangian variables would be added. For the ideal fluid, the total
internal energy functional is
V[q] :=∫
d3a ρ0 U (ρ0/J , s0) . (96)
19
Here we have used the fact that a fluid element carries a specfic entropy s = s0(a) and a mass
determined by ρ = ρ0(a)/J . In Section III C we will describe in detail the map from Lagrangian
to Eulerian variables.
Thus, the special case of the action principle of (79) for the ideal fluid has Lagrange’s La-
grangian L[q, q] = T − V . Variation of this action gives the Lagrangian equation of motion for the
fluid
ρ0qi = −A ji∂
∂a j
(ρ2
0
J2
∂U∂ρ
), (97)
with an additional term that describes the J × B force in Lagrangian variables for MHD. See, e.g.,
Morrison [72, 73], Newcomb [76] for details of this calculation and the MHD extension.
B. Hamiltonian formalism in Lagrangian description
Upon defining the momentum density as usual by
πi =δLδqi = ρ0 qi , (98)
we can obtain the Hamiltonian by Legendre transformation, yielding
H[π,q] = T + V =
∫d3a
(|π|2
2ρ0+ ρ0U (ρ0/J , s0)
), (99)
where |π|2 = πiπi = πiηi jπ j. This Hamiltonian with the canonical Poisson bracket,
{F,G} =
∫d3a
(δFδqi
δGδπi−δGδqi
δFδπi
), (100)
yields
qi = {qi,H} = πi/ρ0 , (101)
πi = {πi,H} = −A ji∂
∂a j
(ρ2
0
J2
∂U∂ρ
). (102)
Equations (101) and (102) are equivalent to (97). For MHD a term HB is added to (99) [see 73, 76].
We will give this explicitly in the constraint context in Section IV B 1 after discussing the Lagrange
to Euler map.
C. Hamiltonian formalism in Eulerian description via the Lagrange to Euler map
In order to understand how constraints in terms of the Lagrangian variable description relate
to those in terms of the Eulerian description, in particular ∇ · v = 0, it is necessary to understand
20
the mapping from Lagrangian to Eulerian variables. Thus, we record here the relationship be-
tween the two unconstrained descriptions, i.e., how the noncanonical Hamiltonian structure of the
compressible Euler’s equations relates to the Hamiltonian structure described in Section III B.
For the ideal fluid, the set of Eulerian variables can be taken to be {v, ρ, s}, where v(x, t) is the
velocity field at the Eulerian observation point, x = (x, y, z) = (x1, x2, x3) at time t and, as as noted
in Section III A, ρ(x, t) is the mass density and s(x, t) is the specific entropy. In order to describe
magnetofluids the magnetic field B(x, t) would be appended to this set. It is most important to
distinguish between the Lagrangian fluid element position and label variables, q and a, and the
Eulerian observation point x, the latter two being independent variables. Confusion exists in the
literature because some authors use the same symbol for the Lagrangian coordinate q and the
Eulerian observation point x.
The Lagrangian and Eulerian descriptions must clearly be related and, indeed, knowing q(a, t)
we can obtain v(x, t). If one were to insert a velocity probe into a fluid at (x, t) then one would
measure the velocity of the fluid element that happened to be at that position at that time. Thus it
is clear that q(a, t) = v(x, t), where recall the overdot means the time derivative at constant a. But,
which fluid element will be at x at time t? Evidently x = q(a, t), which upon inversion yields the
label of that element that will be measured, a = q−1(x, t). Thus, the Eulerian velocity field is given
by
v(x, t) = q(a, t)|a=q−1(x,t) . (103)
Properties can be attached to fluid elements, just as a given mass is identified with a given particle
in mechanics. For a continuum system it is natural to attach a mass density, ρ0(a), to the element
labeled by a. Whence the element of mass in a given volume is given by ρ0d3a and this amount of
mass is preserved by the flow, i.e. ρ(x, t)d3x = ρ0(a)d3a. Because the locus of points of material
surfaces move with the fluid are determined by q, an initial volume element d3a maps into a
volume element d3x at time t according to
d3x = Jd3a , (104)
Thus, using (104) we obtain ρ0 = ρJ as used in Section III A.
Other quantities could be attached to a fluid element; for the ideal fluid, entropy per unit mass,
s(x, t), is such a quantity. The assumption that each fluid element is isentropic then amounts to
s = s0. Similarly, for MHD a magentic field, B0(a), can be attached, and then the frozen flux
assumption yields B · d2x = B0 · d2a. An initial area element d2a maps into an area element d2x at
21
time t according to
(d2x)i = A ji (d2a) j . (105)
Using (105) we obtain JBi = B j0 ∂qi/∂a j.
Sometimes it is convenient to use another set of Eulerian density variables: {M, ρ, σ,B}, where
σ = ρs is the entropy per unit volume, and M = ρv is the momentum density. These Eulerian
variables can be represented by using the Dirac delta function to ‘pluck out’ the fluid element that
happens to be at the Eulerian observation point x at time t. For example, the mass density ρ(x, t)
is obtained by
ρ(x, t) =
∫d3a ρ0(a) δ (x − q (a, t)) =
ρ0
J
∣∣∣∣∣a=q−1(x,t)
. (106)
The density one observes at x at time t will be the one attached to the fluid element that happens
to be there then, and this fluid element has a label given by solving x = q(a, t). The second
equality of (106) is obtained by using the three-dimensional version of the delta function identity
δ( f (x)) =∑
i δ(x − xi)/| f ′(xi)|, where f (xi) = 0. Similarly, the entropy per unit volume is given by
σ(x, t) =
∫d3aσ0(a) δ (x − q (a, t)) =
σ0
J
∣∣∣∣∣a=q−1(x,t)
, (107)
which is consistent with σ0(a) = ρ0(a)s0(a) and s(x, t) = s0(a)|a=q−1(x,t), where the latter means s
is constant along a Lagrangian orbit. Proceeding, the momentum density, M = (M1,M2,M3), is
related to the Lagrangian canonical momentum (defined in Sec. III B) by
M(x, t) =
∫d3aπ(a, t) δ (x − q(a, t)) =
π(a, t)J
∣∣∣∣∣a=q−1(x,t)
, (108)
where for the ideal fluid and MHD, π(a, t) = (π1, π2, π3) = ρ0q. Lastly,
Bi(x, t) =
∫d3a
∂qi(a, t)∂a j B j
0(a) δ (x − q(a, t)) =∂qi(a, t)∂a j
B j0(a)J
∣∣∣∣∣∣∣a=q−1(x,t)
, (109)
for the components of the magnetic field. It may be unfamiliar to view the magnetic field as
density, but in MHD it obeys a conservation law. Geometrically, however, it is naturally a vector
density associated with a differential 2-form.
To obtain the noncanonical Eulerian Poisson bracket we consider functionals F[q, π] that are
restricted so as to obtain their dependence on q and π only through the Eulerian variables. Upon
setting F[q,π] = F[v, ρ, σ], equating variations of both sides,
δF =
∫d3a
[δFδq· δq +
δFδπ· δπ
]=
∫d3x
[δFδρδρ +
δFδσ
δσ +δFδM· δM
]= δF , (110)
22
varying the expressions (106), (107), and (108), substituting the result into (110), and equating the
independent coefficients of δq and δπ, we obtain
δFδq
=
∫d3x
[ρ0 ∇
δFδρ
+ σ0 ∇δFδσ
+ πi∇δFδMi
]δ(x − q) , (111)
δFδπ
=
∫d3x
δFδM
δ(x − q) . (112)
(See [72] and [74] for details.) Upon substitution of (111) and (112), expressions of the functional
chain rule that relate Lagrangian functional derivatives to the Eulerian functional derivates, into
(100) yields the following bracket expressed entirely in terms of the Eulerian fields {M, ρ, σ}:
{F,G} = −
∫d3x
[Mi
(δFδM j
∂
∂x j
δGδMi−δGδM j
∂
∂x j
δFδMi
)+ ρ
(δFδM· ∇
δGδρ−δGδM· ∇
δFδρ
)+ σ
(δFδM· ∇
δGδσ−δGδM· ∇
δFδσ
)]. (113)
In (113) we have dropped the overbars on the Eulerian functional derivatives. The bracket for
MHD is the above with the addition of the following term, which is obtained by adding a B
contribution to (110):
{F,G}B = −
∫d3x
[B ·
(δFδM· ∇
δGδB−δGδM· ∇
δFδB
)+ B ·
(∇
(δFδM
)·δGδB− ∇
(δGδM
)·δFδB
) ], (114)
where dyadic notation is used; for example, B · [∇(D) · C] =∑
i, j BiC j∂D j/∂xi, for vectors B,D,
and C. Alternatively, the bracket in terms of {v, ρ, s,B} is obtained using chain rule expressions,
e.g.,δFδρ
∣∣∣∣∣∣v,s
=δFδρ
∣∣∣∣∣∣M,s
+Mρ·δFδM
+σ
ρ
δFδσ
, (115)
yielding
{F,G} = −
∫d3x
[(δFδρ∇ ·
δGδv−δGδρ∇ ·
δFδv
)+
(∇ × vρ·δGδv×δFδv
)+∇sρ·
(δFδsδGδv−δGδs
δFδv
)], (116)
and
{F,G}B = −
∫d3x
[B ·
(1ρ
δFδv· ∇
δGδB−
1ρ
δGδv· ∇
δFδB
)+ B ·
(∇
(1ρ
δFδv
)·δGδB− ∇
(1ρ
δGδv
)·δFδB
) ]. (117)
23
The bracket of (116) plus that of (117) with the Hamiltonian
H[ρ, s, v,B] =
∫d3x
(12ρ|v|2 + ρU(ρ, s) +
12|B|2
)(118)
gives the Eulerian version of MHD in Hamiltonian form, ∂v/∂t = {v,H}, etc., and similarly using
(113) plus (114) with the Hamiltonian expressed in terms of (M, ρ, σ,B). Ideal fluid follows upon
neglecting the B terms.
D. Constants of motion: Eulerian vs. Lagrangian
In oder to compare the imposition of constraints in the Lagrangian and Eulerian descriptions, it
is necessary to compare Lagrangian and Eulerian conservations laws. This is because constraints,
when enforced, are conserved quantities. The comparison is not trivial because time independent
quantities in the Eulerian description can be time dependent in the Lagrangian description.
Consider a Lagrangian function f (a, t), typical of the Lagrangian variable description, and the
relation x = q(a, t), which relates an Eulerian observation point x to a corresponding fluid element
trajectory value. The function f can be written in either picture by composition, as follows:
f (a, t) = f (x, t) = f (q(a, t), t) , (119)
where we will use a tilde to indicated the Eulerian form of a Lagrangian function. Application of
the chain rule gives
Aik
J
∂ f∂ai
∣∣∣∣∣∣a=q−1(x,t)
=∂ f∂xk and
Ak`
J
∂
∂ak
(π`
J
)∣∣∣∣∣∣a=q−1(x,t)
= ∇ · v , (120)
with the second equality of (120) being a special case of the first. Similarly,
f∣∣∣a=q−1(x,t) =
∂ f∂t
+ qi(a, t)∂ f∂xi
∣∣∣∣∣∣a=q−1(x,t)
=∂ f∂t
+ v · ∇ f (x, t) , (121)
where recall an overdot denotes the time derivative at constant a, ∂/∂t denotes the time derivative
at constant x, and ∇ is the Eulerian gradient with components ∂/∂xi as used in (120). Because
the Jacobian determinant J is composed of derivatives of q, we have J(a, t)|a=q−1(x,t) = J(x, t),
whence one obtains a formula due to Euler [see e.g. 80],
∂J
∂t+ v · ∇J = J ∇ · v . (122)
24
Now, consider a conservation law in the Lagrangian variable description,
DL +∂ΓiDL
∂ai = 0 , (123)
where the densityDL(a, t) has the associated flux ΓDL . Then, the associated conserved quantity is
IDL =
∫d3aDL , (124)
which satisfies dIDL/dt = 0 provided surface terms vanish. Similarly, an Eulerian conservation
law with densityDE and flux ΓDE is
∂DE
∂t+∂ΓiDE
∂xi = 0 (125)
and the following is similarly constant in time:
IDE =
∫d3xDE . (126)
The relationship between the two conservation laws (123) and (125) can be obtained by defining
DL = JDE , ΓiDL
= Aik ΓkDE, and ΓDE = ΓDE + vDE , (127)
and their equivalence follows from (92), (121), and (122). Given a Lagrangian conservation law,
one can use (127) to obtain a corresponding Eulerian conservation law. The densityDE is obtained
from the first equation of (127), a piece of the Eulerian flux ΓDE from the second, which then can
be substitued into the third equation of (127) to obtain the complete Eulerian flux ΓDE . An Eulerian
conservation law is most useful when one can write DE and ΓDE entirely in terms of the Eulerian
variables of the fluid.
The simplest case occurs when DL only depends on a, in which case the corresponding flux
is zero and ∂DL/∂t = 0 and dIDL/dt = 0 follow directly because (124) has no time dependence
whatsoever. Any attribute attached to a fluid element only depends on the label a and this has a
trivial conservation law of this form. However, such trivial Lagrangian conservation laws yield
nontrivial Eulerian conservation laws. Observe, even thought ΓDE ≡ 0 by (127), ΓDE = vDE , 0.
Consider the case of the entropy whereDL = s0(a), whence s(x, t) = s0(a(x, t)) and by (121),
∂s∂t
+ v · ∇s = 0 , (128)
with the quantity s = s0/J being according to (127) the Eulerian conserved density, as can be
verified using (122). But, as it stands, this density cannot be written in terms of Eulerian fluid
25
variables. However, σ0 = ρ0s0 is also a trivial Lagrangian conserved density and according to
(127) we have the Eulerian density ρ0s0/J = ρs = σ that satisfies
∂σ
∂t+ ∇ · (vσ) = 0 . (129)
Thus, it follows that any advected scalar has an associated conserved quantity obtained by multi-
plication by ρ.
As another example, consider the quantity Bi0∂q j/∂ai. This quantity is the limit displacement
between two nearby fluid elements, i.e., q(a, t) − q(a + δa, t) along the initial magnetic field as
δa→ 0. Evidently,˙(
Bi0∂q j
∂ai
)= Bi
0∂q j
∂ai =∂
∂ai
(Bi
0q j), (130)
where the second equality follows if the initial magnetic field is divergence free. This is of course
another trivial conservation law, for the time derivative of a density that is a divergence will always
be a divergence. However, let us see what this becomes in the Eulerian description. According
to (127) the corresponding Eulerian density is DE = DL/J ; so, the density associated with this
trivial conservation law (130) is
B j(x, t) =Bi
0
J
∂q j
∂ai
∣∣∣∣∣∣a=q−1(x,t)
. (131)
which as we saw in Section III C is the expression one gets for the MHD magnetic field because
of flux conservation. That the divergence-free magnetic field satisfies a conservation law is clear
from∂B∂t
= −v · ∇B + B · ∇v − B∇ · v = ∇ ·↔
T , (132)
where the tensor↔
T of the last equality is
↔
T = B ⊗ v − v ⊗ B . (133)
Thus we have another instance where a trivial Lagrangian conservation law leads to a nontrivial
Eulerian one.
Although Bi0∂q j/∂ai does not map into an expression entirely in terms of our set of Eulerian
variables, we can divide it by ρ0, a quantity that only depends on the label a, and obtain
Bi0
ρ0
∂q j
∂ai
∣∣∣∣∣∣a=q−1(x,t)
=B j
ρ. (134)
26
Eulerianizing the counterpart of (130) for this expression gives
∂
∂t
(Bρ
)+ v · ∇
(Bρ
)=
Bρ· ∇v , (135)
which is not an Eulerian conservation law. This is to be expected because, unlike what we did to get
(132), we have Eulerianized without using (127). In light of its relationship to q(a, t)−q(a +δa, t),
the quantity B/ρ has been described as a measure of the distance of points on a magnetic field line
[see e.g. 68]. This was predated by analogous arguments for vorticity [see e.g. 80].
IV. CONSTRAINT THEORIES FOR THE INCOMPRESSIBLE IDEAL FLUID
A. The incompressible fluid in Lagrangian variables
In order to enforce incompressibility, Lagrange added to his Lagrangian the constraint J = 1
with the Lagrange multiplier λ(a, t),
Lλ[q, q] = T [q] + λJ , (136)
with T given (95). Here we have dropped V because incompressible fluids contain no internal
energy. Upon insertion of (136) into the action of Hamilton’s principle it is discovered that λ
corresponds to the pressure. The essence of this procedure was known to Lagrange. (See Serrin
[80] for historical details and Sommerfeld [81] for an elementary exposition.) This procedure
yields
ρ0qi = −Aij∂λ
∂a j , (137)
where use has been made of (92). The Eulerian form of (137) is clearly ρ(∂v/∂t + v · ∇v) = −∇λ,
whence it is clear that λ is the pressure. Although Lagrange knew the Lagrange multiplier was the
pressure, he could only solve for it in special cases. The general procedure of Section I A was not
available because Green’s function techniques and the theory of elliptic equations were not at his
disposal.
1. Lagrangian volume preserving geodesic flow
If the constraint is dropped from (136), we obtain free particle motion for an infinite-dimensional
system, the ideal fluid case of (85) of Section II D, which is analogous to the finite-dimensional
27
case of Section II A 1. Because the constraint J = 1 only depends on the derivatives of q, it is a
configuration space constraint; thus, it is an holonomic constraint. As is well-known and reviewed
in Section II A 1, free particle motion with holonomic constaints is geodesic flow. Thus, following
Lagrange, it is immediate that the ideal incompressible fluid is an infinite-dimensional version of
geodesic flow.
Lagrange’s calculation was placed in a geometric/group theoretic setting in [50] (see also Ap-
pendix 2 of [51] and [52]). Given that the transformation a ↔ q, at any time, is assumed to be
a smooth invertible coordinate change, it is a Lie group, one referred to as the diffeomorphism
group. With the additional assumption that these transformations are volume preserving, La-
grange’s constraint J = 1, the transformations form a subgroup, the group of volume preserving
diffeomorphisms. Thus, Lagrange’s work can be viewed as geodesic flow on the group of volume
preserving diffeomorphisms.
Although Arnold’s assumptions of smoothness etc. are mathematically dramatic, his descrip-
tion of Lagrange’s calculations in these terms has spawned a considerable body of research. Asso-
ciated with a geodesic flow is a metric, and whence one can calculate a curvature. In his original
work, Arnold added the novel calculation of the curvature in the mathematically more forgiving
case of two-dimensional flow with periodic boundary conditions.
B. Lagrangian-Dirac constraint theory
More recently there have been several works [55, 75, 84], following Nguyen & Turski [77, 78],
that treat the enforcement of the incompressibility constraint of hydrodynamics by Dirac’s method
of constraints [62]. In these works the compressibility constraint was enforced in the Eulerian
variable description of the fluid using the noncanonical Poisson bracket of Section III C as the base
bracket of a generalization of Dirac’s constraint theory. We will return to this approach in Section
III C where we revisit and extend Dirac’s constraint results for the fluid in the Eulerian variable
description. Here, apparently for the first time, we consider the incompressibility constraint in the
Lagrangian variable description, where the canonical Poisson bracket of (100) is the base for the
construction of a Dirac bracket.
We adapt (84) for the fluid case at hand with the supposition of only two local constraints,
which we write as
Da(a′) =
∫d3a Da(a) δ(a − a′) , (138)
28
where a = 1, 2 and Da(a) is a shorthand for a function of q(a, t) and π(a, t) and their derivatives
with respect to a. Then the matrix D is a 2 × 2 matrix with the components
Dab(a, a′) = {Da(a),Db(a′)} ,
using the canonical bracket of (100). To construct the Dirac bracket
{F,G}∗ = {F,G} −∫
d3a∫
d3a′ {F,Da(a)}D−1ab (a, a′){Db(a′),G}, (139)
we require the inverse, which satisfies∫d3a Dac(a′, a)D−1
cb (a, a′′) = δab δ(a
′ − a′′) . (140)
Rather than continuing with the general case, which is unwieldy, we proceed to the special
case for the incompressible fluid, an infinite-dimensional version of the holonomic constraints
discussed in Section II C 1.
1. Lagrangian-Dirac incompressibility holonomic constraint
Evidently we will want our holonomic incompressibility constraint to be J . However, it is
convenient to express this by choosing
D1 = ln(J
ρ0
). (141)
This amounts to the same constraint as J = 1 with the value D1 = − ln(ρ0). The scaling of J
in (141) by ρ0(a) is immaterial because it is a time-independent quantity. To obtain the second
constraint we follow suit and set
D2 = D1 =Ak`
J
∂
∂ak
(π`
ρ0
)= η` j Ak
`
J
∂
∂ak
(π j
ρ0
), (142)
where recall we assume η` j = δ` j and π j is given by (98). That the constraint D2 is the time
derivative of D1 requires the definition of π j of (98) that uses the Hamiltonian∫
d3a |π|2/(2ρ0).
Observe, that constraints D1 and D2 are local constraints in that they are enforced pointwise [see
e.g. 65], i.e., they are enforced on each fluid element labeled by a. Equation (141) corresponds in
the Eulerian picture to − ln(ρ), while the second constraint of (142), the Lagrangian time derivative
of the first constraint, corresponds in the Eulerian picture to ∇ · v, which can be easily verified
using the second equation of (120). Note, the particular values of these constraints of interest are,
29
of course, J = 1 and ∇ · v = 0, but the dynamics the Dirac bracket generates will preserve any
values of these constraints. For example, we could set J = f (a) where the arbitrary function f is
less than unity for some a and greater for others, corresponding to regions of fluid elements that
experience contraction and expansion. Also note, because we have used π with the up index in
(142); thus as seen in the second equality it depends on the metric. This was done to make it have
the Eulerian form ∇ · v.
For the constraints (141) and (142), D only depends on two quantities because D1 does not
depend on π, i.e. {D1,D1} = 0 and {D1,D2} = −{D2,D1}. Thus the inverse has the form
D−1 =
D−111 D
−112
D−121 0
, (143)
giving rise to the conditions
D−112 · D
21 = I = D−121 · D
12 and D−111 · D
12 + D−112 · D
22 = 0 , (144)
where I is the identity. Thus, the inverse is easily tractable if the inverse of D12 exists; whence,
D−111 = −D−1
12 · D22 · D−1
21 . (145)
In the above the symbol ‘ · ’ is used to denote the product with the sum in infinite dimensions,
i.e., integration over d3a as in (140). Equation (145) can be rewritten in an abbreviated form with
implied integrals on repeated arguments as
D−111 (a′, a′′) = D−1
21 (a′, a) · D22(a, a) · D−121 (a, a′′) . (146)
In order to obtain D and its inverse, we need the functional derivatives of D1 and D2. These are
obtained directly by writing these local constraints as in (140), yielding
δD1(a′)δqi(a)
= −Aki∂
∂ak
δ(a − a′)J
, (147)
δD1(a′)δπi(a)
= 0 , (148)
where use has been made of (92), and
δD2(a′)δqi(a)
=∂
∂au
(Ak
i Au`
J2
∂
∂ak
(π`
ρ0
)δ(a − a′)
), (149)
δD2 (a′)δπi(a)
= −ηi j
ρ0
∂
∂am
(Amj
Jδ(a − a′
)), (150)
30
where use has been made of (93) and recalling we have (91) at our disposal.
Let us now insert (147), (148), (149), and (150) into the canonical Poisson bracket (100), to
obtain
D12(a, a′) = {D1(a),D2(a′)}
= −A`
i
J
∂
∂a`
(ηi j
ρ0Ak
j∂
∂ak
(δ(a − a′)J
)), (151)
which corresponds to the symmetric matrix S of (34) and (49) and
D22(a, a′) = {D2(a),D2(a′)}
=Ak
i Au`
J2
∂
∂ak
(π`
ρ0
)∂
∂au
[ηi j
ρ0
∂
∂am
(Amj
Jδ(a − a′
))]−
Ami
J
∂
∂am
ηi j
ρ0
∂
∂au
AkjA
u`
J2
∂
∂ak
(π`
ρ0
)δ(a − a′)
, (152)
which corresponds to the antisymmetric matrix A of (35). Observe the symmetries corresponding
to the matrices S and A, respectively, are here∫d3a′D12(a, a′) φ(a′) =
∫d3a′D12(a′, a) φ(a′) ,∫
d3a′D22(a, a′) φ(a′) = −
∫d3a′D22(a′, a) φ(a′) ,
for all functions φ. The first follows from integration by parts, while the second is obvious from
its definition.
Using (140), the first condition of (144) is∫d3a′′ D12(a′, a′′)D−1
21 (a′′, a) = δ(a′ − a) , (153)
which upon substitution of (151) and integration gives
−A`
i
J
∂
∂a`
[ηi j
ρ0Ak
j∂
∂ak
(D−1
21 (a, a′′)J
)]= δ(a − a′′) . (154)
We introduce the formally self-adjoint operator (cf. (91))
∆ρ0 f =A`
i
J
∂
∂a`
[ηi j
ρ0Ak
j∂
∂ak
(fJ
)], (155)
i.e., an operator that satisfies∫d3a f (a) ∆ρ0g(a) =
∫d3a g(a) ∆ρ0 f (a) , (156)
31
a property inherited by its inverse ∆−1ρ0
. Thus we can rewrite equation (154) as
D−121 (a, a′′) = −G0
(a, a′′
)= −∆−1
ρ0δ(a − a′′) , (157)
where G0 represents the Green function associated with (154).
In order to obtain D−121 , we find it convenient to transform (157) to Eulerian variables. Using
x = q(a, t) we findD−1
21 (a, a′)J
= −G(x, x′) = −G(q(a),q(a′)) , (158)
where G satisfies
∇ ·
(1ρ∇G
)= −∆ρ
D−121 (a, a′)J
= Jδ(x − x′). (159)
Here use has been made of identities (91) and (120). As noted in Section I A, under physically
reasonable conditions, the operator
∆ρ f = ∆ρ0 (J f ) = ∇ ·
(1ρ∇ f
)(160)
has an inverse. Thus we write
D−121 (a, a′) = −J∆−1
ρ
(J δ
(q(a, t) − q(a′, t)
)). (161)
Now, using D−121 = −D−1
12 , the element D−111 follows directly from (145).
For convenience we write the Dirac bracket of (139) as follows:
{F,G}∗ = {F,G} − [F,G]D , (162)
where
[F,G]D :=2∑
a,b=1
[F,G]Dab =
∫d3a
∫d3a′ {F,Da (a)}D−1
ab (a, a′){Db (
a′),G
}. (163)
Because D−122 = 0 and [F,G]D
12 = −[G, F]D21, we only need to calculate [F,G]D
11 and [F,G]D21.
As above, we substitute (147), (148), (149), and (150) into the bracket (100) and obtain{F,D1 (a)
}= −
Aki
J
∂
∂ak
(δFδπi
), (164)
{F,D2 (a)
}=
Ak`
J
∂
∂ak
(ηi`
ρ0
δFδqi
)+
Aki
J
Au`
J
∂
∂ak
(π`
ρ0
)∂
∂au
(δFδπi
). (165)
Then, exploiting the antisymmetry of the Poisson bracket, it is straightforward to calculate analo-
gous expressions for the terms{D1,2,G
}.
32
We first analyze the operator
[F,G]D11 =
∫d3a
∫d3a′
∫d3a
∫d3a
{F,D1(a)
}D−1
21 (a, a)D22(a, a)D−121 (a, a′)
{D1(a′) ,G}
, (166)
where we used the second condition of (144) to replace D−111 . Upon inserting (161) and (164), this
equation can be rewritten as
[F,G]D11 = −
∫d3a
∫d3a′
∫d3a
∫d3aAh
j
J
∂
∂ah
(δFδπ j
)∆−1ρ0δ(a − a)
a=a
D22(a, a)[As
r
J
∂
∂as
(δGδπr
)∆−1ρ0δ(a − a)
]a=a′
, (167)
where the subscripts on the right delimiters indicate that a is to be replaced after the derivative
operations, including those that occur in J and A ji .
Integrating this expression by parts with respect to a and a′ yields
[F,G]D11 = −
∫d3a
∫d3a
∆−1ρ0
Ahj
J
∂
∂ah
(δFδπ j
)a=a
D22(a, a)[∆−1ρ0
(As
r
J
∂
∂as
(δGδπr
))]a=a
, (168)
and then substituting (152) gives
[F,G]D11 = −
∫d3a
∫d3a
∆−1ρ0
Ahj
J
∂
∂ah
(δFδπ j
)a=a
{Ak
i Au`
J2
∂
∂ak
(π`
ρ0
)∂
∂au
[ηin
ρ0
∂
∂am
(Am
n
Jδ (a − a)
)]−
Ami
J
∂
∂am
[ηin
ρ0
∂
∂au
(Ak
n
J
Au`
J
∂
∂ak
(π`
ρ0
)δ(a − a)
)]}a=a
[∆−1ρ0
(As
r
J
∂
∂as
(δGδπr
))]a=a
. (169)
Then, by means of integrations by parts we can remove the derivatives from the term δ(a − a) and
perform the integral. After relabeling the integration variable as a to simplify the notation, (169)
becomes
[F,G]D11 =
∫d3a
{ρ0 η
ui Aku
J
∂
∂ak
(π`
ρ0
) (Pρ0⊥
δFδπ
∣∣∣∣∣`
Pρ0⊥
δGδπ
∣∣∣∣∣i− Pρ0⊥
δFδπ
∣∣∣∣∣iPρ0⊥
δGδπ
∣∣∣∣∣`
)
+ ηniAu`
∂
∂au
[Ak
n
J
∂
∂ak
(π`
ρ0
)] Pρ0⊥
δGδπ
∣∣∣∣∣i
∆−1ρ0
J
Ahj
J
∂
∂ah
(δFδπ j
)− Pρ0⊥
δFδπ
∣∣∣∣∣i
∆−1ρ0
J
Ahj
J
∂
∂ah
(δGδπ j
)}, (170)
where we introduced the projection operator
(Pρ0⊥
)i
jz j =
ηi`
ρ0Au`
∂
∂au
∆−1ρ0
J
Ahj
J
∂
∂ah z j
=: Pρ0⊥z∣∣∣i , (171)
33
where in the last equality we defined a shorthand for convenience; similarly we use
Pρ0⊥z∣∣∣`
:=1ρ0
Au`
∂
∂au
∆−1ρ0
J
Ahj
J
∂
∂ah z j
. (172)
It is straightforward to prove that Pρ0⊥ represents a projection, i.e. Pρ0⊥
(Pρ0⊥z
)= Pρ0⊥z for each
z, which in terms of indices would have an ith component given by (Pρ0⊥)ij(Pρ0⊥) j
k zk = (Pρ0⊥)ik zk.
Also, Pρ0⊥ is formally self-adjoint with respect to the following weighted inner product:∫d3a ρ0 wi (Pρ0⊥)i
j z j =
∫d3a ρ0 zi (Pρ0⊥)i
j w j . (173)
The projection operator complementary to Pρ0⊥ is given by
Pρ0 = I − Pρ0⊥ , (174)
where I is the identity.
Now let us return to our evaluation of [F,G]D and analyze the contribution
[F,G]D21 =
∫d3a
∫d3a′
{F,D2 (a)
}D−1
21 (a, a′){D1 (
a′),G
}. (175)
Using (161), (164), and (165), this equation can be rewritten as
[F,G]D21 = −
∫d3a
∫d3a′
[Ak
i
J
∂
∂ak
(ηin
ρ0
δFδqn
)+
Aki
J
Au`
J
∂
∂ak
(π`
ρ0
)∂
∂au
(δFδπi
)]× ∆−1
ρ0δ(a − a′)
Ahj
J
∂
∂ah
(δGδπ j
)a=a′
(176)
and, integrating by parts to simplify the δ(a − a′) term, results in
[F,G]D21 =
∫d3a
{δFδqi Pρ0⊥
δGδπ
∣∣∣∣∣i + ρ0Ak
i
J
∂
∂ak
(π`
ρ0
)δFδπiPρ0⊥
δGδπ
∣∣∣∣∣`
+ Au`
∂
∂au
[Ak
i
J
∂
∂ak
(π`
ρ0
)]δFδπi
∆−1ρ0
J
Ahj
J
∂
∂ah
(δGδπ j
)}. (177)
We can now combine the operators [F,G]D11, [F,G]D
21, and [F,G]D12 = −[G, F]D
21, given by (170)
and (177), to calculate the Dirac bracket (162). First, we rewrite (163) as
[F,G]D =
∫d3a
{δFδqi Pρ0⊥
δGδπ
∣∣∣∣∣i − δGδqi Pρ0⊥
δFδπ
∣∣∣∣∣i+ ρ0
Aki
J
∂
∂ak
(π`
ρ0
) (Pρ0
δFδπ
∣∣∣∣∣i Pρ0⊥
δGδπ
∣∣∣∣∣`
− Pρ0
δGδπ
∣∣∣∣∣i Pρ0⊥
δFδπ
∣∣∣∣∣`
)+ Au
`
∂
∂au
[Ak
i
J
∂
∂ak
(π`
ρ0
)] Pρ0
δFδπ
∣∣∣∣∣i ∆−1ρ0
J
Ahj
J
∂
∂ah
(δGδπ j
)− Pρ0
δGδπ
∣∣∣∣∣i ∆−1ρ0
J
Ahj
J
∂
∂ah
(δFδπ j
)}. (178)
34
Using the identity of (94) with z` set to π`/ρ0,
Au`
∂
∂au
[Ak
i
J
∂
∂ak
(π`
ρ0
)]= Ak
i∂
∂ak
[Au`
J
∂
∂au
(π`
ρ0
)], (179)
and integrating by parts, (178) becomes
[F,G]D =
∫d3a
{δFδqi Pρ0⊥
δGδπ
∣∣∣∣∣i − δGδqi Pρ0⊥
δFδπ
∣∣∣∣∣i+ ρ0
Aki
J
∂
∂ak
(π`
ρ0
) (Pρ0
δFδπ
∣∣∣∣∣i Pρ0⊥
δGδπ
∣∣∣∣∣`
− Pρ0
δGδπ
∣∣∣∣∣i Pρ0⊥
δFδπ
∣∣∣∣∣`
)− ρ0
Au`
J
∂
∂au
(π`
ρ0
) (Pρ0
δFδπ
∣∣∣∣∣i Pρ0⊥
δGδπ
∣∣∣∣∣i− Pρ0
δGδπ
∣∣∣∣∣i Pρ0⊥
δFδπ
∣∣∣∣∣i
)}, (180)
where we used
Aki∂
∂akPρ0z∣∣∣i = Ak
i∂
∂ak
(Pρ0
)i
jz j = 0 , for all z , (181)
which follows from the definitions (171), viz.
Aki
J
∂
∂ak
(Pρ0⊥
)i
jz j =
Aki
J
∂z j
∂ak , (182)
and (174). Also, upon inserting Pρ0⊥ = I − Pρ0 in the last line of (180), symmetry implies we can
drop the Pρ0⊥. Finally, upon substituting (180) into (162), we obtain
{F,G}∗ =
∫d3a
{δFδqi Pρ0
δGδπ
∣∣∣∣∣i − δGδqi Pρ0
δFδπ
∣∣∣∣∣i− ρ0
Aki
J
∂
∂ak
(π`
ρ0
) (Pρ0
δFδπ
∣∣∣∣∣i Pρ0⊥
δGδπ
∣∣∣∣∣`
− Pρ0
δGδπ
∣∣∣∣∣i Pρ0⊥
δFδπ
∣∣∣∣∣`
)+ ρ0
Ak`
J
∂
∂ak
(π`
ρ0
) (Pρ0
δFδπ
∣∣∣∣∣i δGδπi − Pρ0
δGδπ
∣∣∣∣∣i δFδπi
)}. (183)
Once more inserting Pρ0⊥ = I − Pρ0 , rearranging, and reindexing gives
{F,G}∗ = −
∫d3a ρ0
{1ρ0
δGδqi Pρ0
δFδπ
∣∣∣∣∣i − 1ρ0
δFδqi Pρ0
δGδπ
∣∣∣∣∣i +Amn Pρ0⊥
δFδπ
∣∣∣∣∣m Pρ0⊥
δGδπ
∣∣∣∣∣n+ Tmn
(δFδπmPρ0⊥
δGδπ
∣∣∣∣∣n − δGδπmPρ0⊥
δFδπ
∣∣∣∣∣n)}, (184)
where
Anm := η`mD`n − η`nD`
m and Tmn := η`nD`m + ηmnD2 , (185)
35
with
D`m =
Akm
J
∂
∂ak
(π`
ρ0
). (186)
Note the trace D`` = D2, which we will eventually set to zero. Equation (184) gives the Dirac
bracket for the incompressibility holonomic constraint. This bracket with the Hamiltonian
H =
∫d3a|π|2
2ρ0=
∫d3a ηmnπmπn
2ρ0, (187)
produces dynamics that fixesJ and thus enforces incompressibility provided the constraint D2 = 0
is used as an initial condition. For MHD we add to H the following:
HB =
∫d3a ηmn
B j0Bk
0
2J∂qm
∂a j
∂qn
∂ak . (188)
We note, any Hamiltonian that is consistent with (142) can be used to define a constrained flow.
Proceeding to the equations of motion, we first calculate qi,
qi = {qi,H}∗ =(Pρ0
)i
j
δHδπ j
=δHδπi−ηi`
ρ0Au`
∂
∂au
∆−1ρ0
J
Ahj
J
∂
∂ah
δHδπ j
=πi
ρ0−ηi`
ρ0Au`
∂
∂au
∆−1ρ0
J
Ahj
J
∂
∂ah
π j
ρ0
. (189)
The equation for πi is more involved. Using the adjoint property of (173), which is valid for both
Pρ0⊥ and Pρ0 , we obtain
πi = {πi,H}∗ = −ρ0
(Pρ0
) j
i
1ρ0
δHδq j − ρ0
(Pρ0⊥
)m
i
(Amn
(Pρ0⊥
)n
k
δHδπk
)+ ρ0
(Pρ0⊥
)n
i
(Tmn
δHδπm
)− ρ0 Tin
(Pρ0⊥
)n
k
δHδπk
= −ρ0
(Pρ0⊥
)m
i
(Amn
(Pρ0⊥
)n
k
πk
ρ0
)+ ρ0
(Pρ0⊥
)n
i
(Tmn
πm
ρ0
)− ρ0 Tin
(Pρ0⊥
)n
k
πk
ρ0, (190)
which upon substitution of the definitions of Pρ0 , Amn, and Tmn of (171) and (185) yields a com-
plicated nonlinear equation.
Equations (189) and (190) are infinite-dimensional versions of the finite-dimensional systems
of (40) and (41) considered in Section II C 1. There, equations (40) and (41) were reduced to (44)
and (45) upon enforcing the holomomic constraint by requiring that initially D2 = 0. Similarly
we can enforce the vanishing of D2 of (142), which is compatible with the Hamiltonian (187).
Instead of addressing this evaluation now, we find the meaning of various terms is much more
transparent when written in terms of Eulerian variables, which we do in Section IV C. We then
36
return to these Lagrangian equations in Section IV D and make comparisons. Nevertheless, the
solution of equations (189) and (190), q(a, t), with the initial conditions D1 = − ln ρ0 and D2 = 0,
is a volume preserving transformation at any time t.
C. Eulerian-Dirac constraint theory
Because we chose the form of constraints D1,2 of (141) and (142) to be Eulerianizable, it follows
that we can transform easily the results of Section IV B 1 into Eulerian form. This we do in Section
IV C 1. Alternatively, we can proceed as in Chandre et al. [55], Morrison et al. [75], Nguyen &
Turski [77, 78], Tassi et al. [84], starting from the Eulerian noncanonical theory of Section III C
and directly construct a Dirac bracket with Eulerian constraints. This is a valid procedure because
Dirac’s construction works for noncanonical Poisson brackets, as shown, e.g., in Morrison et al.
[75], but it does not readily allow for advected density. This direct method with uniform density is
reviewed in Section IV C 2, where it is contrasted with the results of Section IV C 1.
1. Lagrangian-Dirac constraint theory in the Eulerian picture
In a manner similar to that used to obtain (111) and (112), we find the functional derivatives
transform asδFδπi
=1ρ
δFδ3i,
1ρ0
δFδqi =
∂
∂xi
δFδρ−
1ρ
δFδs
∂s∂xi −
1ρ
δFδ3`
∂3`∂xi , (191)
where the expressions on the left of each equality are clearly Lagrangian variable quantities, while
on the right they are Eulerian quantities represented in terms of Lagrangian variables. Substituting
these expressions into (84) and dropping the bar on F and G gives the following bracket in terms
of the Eulerian variables:
{F,G}∗ = −
∫d3x
{ (PρδFδv
∣∣∣∣∣i ∂
∂xi
δGδρ− Pρ
δGδv
∣∣∣∣∣i ∂
∂xi
δFδρ
)+
1ρ
∂s∂xi
(δFδsPρδGδv
∣∣∣∣∣i − δGδsPρδFδv
∣∣∣∣∣i)+
1ρ
∂3`
∂xi
(PρδFδv
∣∣∣∣∣`
PρδGδv
∣∣∣∣∣i − Pρ δGδv∣∣∣∣∣`
PρδFδv
∣∣∣∣∣i)+
1ρ
∂3`
∂x`
(PρδFδv
∣∣∣∣∣i δGδ3i − Pρ δGδv∣∣∣∣∣i δFδ3i
)}, (192)
37
where we used the relations (104) and (120) and we introduced the Eulerian projection operator
PρδFδv
∣∣∣∣∣i = (Pρ)ijδFδ3 j
= ρPρ0
δFδπ
∣∣∣∣∣i and PρδFδv
∣∣∣∣∣i= ηi j Pρ
δFδv
∣∣∣∣∣ j
, (193)
with (Pρ
)i
jz j = δi
j − ηik ∂
∂xk
[∆−1ρ
∂
∂x j
(z j
ρ
)], (194)
which is easily seen to satisfy (Pρ)ij(Pρ)
jk = (Pρ)i
k. Observe, like its Lagrangian counterpart, Pρ is
formally self-adjoint; however, this time we found it convenient to define the projection in such a
way that the self-adjointness is with respect to a different weighted inner product, viz.∫d3xρ
wi (Pρ)ij z j =
∫d3xρ
zi (Pρ)ij w j . (195)
In terms of usual cartesian vector notation
PρδGδv
=δGδv− ∇∆−1
ρ ∇ ·
(1ρ
δGδv
). (196)
Upon writing Pρ = I − Pρ⊥ and decomposing an arbitrary vector field as
z = −∇Φ + ρ∇ × A,
this projection operator yields the component Pρz = ρ∇ × A. Hence, ∇ρ × A = 0 is sufficient for
the operator to be a projection into the space of incompressible vector fields. For convenience we
introduce the associated projector
Pρv := v −1ρ∇∆−1
ρ ∇ · v =1ρPρ(ρv) , (197)
which has the desirable property
∇ · (Pρv) = 0 ∀ v compared to ∇ ·
(1ρPρw
)= 0 ∀ w . (198)
Upon writing Pρ = I − Pρ⊥ and decomposing an arbitrary vector field v as
v = −1ρ∇Φ + ∇ × A,
this projection operator yields the component Pρv = ∇ × A, while Pρ⊥v = ∇Φ/ρ. Note, Pρ is the
Eulerianization of Pρ0 and it is not difficult to write (199) in terms of this quantity.
38
Upon adopting this usual vector notation, the bracket (192) can also be written as
{F,G}∗ = −
∫d3x
[∇δGδρ· Pρ
δFδv− ∇
δFδρ· Pρ
δGδv
+∇sρ·
(δFδsPρδGδv−δGδsPρδFδv
)+∇ × vρ·
(PρδGδv× Pρ
δFδv
)+∇ · vρ
(δFδv· Pρ
δGδv−δGδv· Pρ
δFδv
)]. (199)
For MHD there is a magnetic field contribution to (191) and following the steps that lead to (199)
we obtain
{F,G}∗B = −
∫d3x
[B ·
(1ρPρδFδv· ∇
δGδB−
1ρPρδGδv· ∇
δFδB
)+ B ·
(∇
(1ρPρδFδv
)·δGδB− ∇
(1ρPρδGδv
)·δFδB
) ]. (200)
With the exception of the last term of (199) proportional to ∇ · v and the presence of the Eulerian
projection operator Pρ, (199) added to (200) is identical to the noncanonical Poisson bracket for
the ideal fluid and MHD as given in Morrison & Greene [74]. By construction, we know that (199)
satisfies the Jacobi identity – this follows because it was obtained by Eulerianizing the canonical
Dirac bracket in terms of Lagrangian variables. Guessing the bracket and proving Jacobi for (199)
directly would be a difficult chore, giving credence to the path we have followed in obtaining it.
To summarize, the bracket of (199) together with the Hamiltonian
H =12
∫d3x ρ |v|2 , (201)
the Eulerian counterpart of (187), generates dynamics that can preserve the constraint ∇ · v = 0. If
we add HB =∫
d3x |B|2/2 to (201) and add (200) to (199), then we obtain incompressible MHD.
The fluid case is the Eulerian counterpart of the volume preserving geodesic flow, described orig-
inally by Lagrange in Lagrange variables. Upon performing a series of straightforward manipula-
39
tions, we obtain the following equations of motion for the flow:
∂ρ
∂t= {ρ,H}∗ = −∇ · Pρ
δHδv
= −∇ρ · Pρv , (202)
∂s∂t
= {s,H}∗ = −∇sρ· Pρ
δHδv
= −∇s · Pρv , (203)
∂v∂t
= {v,H}∗ = −1ρPρ
(ρ∇
δHδρ
)+
1ρPρ
(∇sδHδs
)−
1ρPρ
((∇ × v) × Pρ
δHδv
)−∇ · vρPρδHδv
+1ρPρ
(∇ · v
δHδv
)= Pρ∇
|v|2
2− Pρ
((∇ × v) × Pρv
)− (∇ · v)Pρv + Pρ (v∇ · v) . (204)
If we include HB we obtain additional terms to (204) generated by (200) for the projected J × B
force. Observe, equation (204) is not yet evaluated on the constraint D2 = 0, which in Eulerian
variables is ∇ · v = 0. As noted at the end of Section IV B, we turn to this task in Section IV D.
2. Eulerian-Dirac constraint theory direct with uniform density
For completness we recall the simpler case where the Eulerian density ρ is uniformly constant,
which without loss of generality can be scaled to unity. This case was considered in Chandre et al.
[55, 56], Nguyen & Turski [77, 78] (although a trick of using entropy as density was employed in
Chandre et al. [55] to treat density advection). In these works the Dirac constraints were chosen
to be the pointwise Eulerian quantities
D1 = ρ and D2 = ∇ · v , (205)
and the Dirac procedure was effected on the purely Eulerian level. This led to the projector
P := Pρ=1 = 1 − ∇∆−1∇ · , (206)
where ∆ = ∆ρ=1, and the following Dirac bracket:
{F,G}∗ = −
∫d3x
[∇sρ·
(δFδsPδGδv−δGδsPδFδv
)−∇ × vρ·
(PδFδv× P
δGδv
) ]. (207)
Incompressible MHD with constant density is generated by adding the following to (207)
{F,G}∗B = −
∫d3x
[Bρ·
(PδFδv· ∇
δGδB− P
δGδv· ∇
δFδB
)+B ·
(∇
(1ρPδFδv
)·δGδB− ∇
(1ρPδGδv
)·δFδB
) ], (208)
40
and adding |B|2/2 to the integrand of (201).
The bracket of (207) differs from that of (199) in two ways: the projector Pρ is replaced by the
simpler projector P and it is missing the term proportional to ∇ ·v. Given that ∇ ·v cannot be set to
zero until after the equations of motion are obtained, this term gives rise to a significant differences
between the constant and nonconstant density Poisson brackets and incompressible dynamics.
D. Comparison of the Eulerian-Dirac and Lagrangian-Dirac constrained theories
Let us now discuss equations (202), (203) and (204). Given that ∇·Pρv = 0 (cf. (198)) it is clear
that the density and entropy are advected by the incompressible velocity field Pρv, as expected.
However, the meaning of (204) remains to be clarified. To this end we take the divergence of (204)
and again use (198) to obtain
∂(∇ · v)∂t
= −∇ ·(∇ · vPρv
)= −
(Pρv
)·∇ (∇ · v) . (209)
Thus ∇ · v itself is advected by an incompressible velocity field. As with any advection equation,
if initially ∇ · v = 0 , it will remain uniformly zero. After setting ∇ · v = 0 in equation (204) it
collapses down to∂v∂t
= −Pρ (v · ∇v) ; (210)
this is the anticipated equation of motion, the momentum equation of (1) with the insertion of the
pressure given by (5).
Given the discussion of Lagrangian vs. Eulerian constants of motion of Section (III D), that ∇·v
is advected rather than pointwise conserved is to be expected. Our development began with the
constraints D1,2 of (141) and (142) both of which are pointwise conserved by the Dirac procedure,
i.e. DL ≡ 0. This means their corresponding fluxes are identically zero, i.e., in (123) we have
ΓDL ≡ 0 for each. Thus the flux component ΓDE of (127) vanishes and the Eulerian flux for both
D1 and D2 have the form vDE. Because D1 and D2 Eulerianize to − ln(ρ) and ∇ · v, respectively,
we expected equations of the from of (209) for both. We will see in Section IV E that the equation
for D1 in fact follows also because the constraints are Casimir invariants.
Let us return to (190) and compare with the results of Section II C 1. Because the incompress-
ibility condition is an holonomic constraint and Section II C 1 concerns holonomic constraints for
the uncoupled N-body problem, both results are geodesic flows. In fact, one can think of the fluid
case as a continuum version of that of Section II C 1 with an infinity of holonomic constraints– thus
41
we expect similarities between these results. However, because the incompressibility constraints
are pointwise constraints, the comparison is not as straightforward as it would be for global con-
straints of the fluid.
To make the comparison we first observe that the termAmn of (184) must correspond to the term↔
Ai j of (54), since their origin follows an analogous path in the derivation, both are antisymmetric,
and both project from both the left and the right. The analog of (56) according to (171) is
(Pρ0⊥
)i
j
π j
ρ0=ηi`
ρ0Au`
∂
∂au
∆−1ρ0
J
Ahj
J
∂
∂ah
π j
ρ0
=ηi`
ρ0Au`
∂
∂au
∆−1ρ0
J
(D2
) ≡ 0 , (211)
when evaluated on D2 = 0. Unlike (56) a sum, which would here be an integral over d3a, does
not occur because the constraint D2 is a pointwise constraint as opposed to a global constraint.
Also, because the constraints are pointwise, the↔
Ti j is analogous to the terms with Tmn that also
have a factor of the projector Pρ0⊥, giving the results analogous to (57). Just as in Section II C 1,
we obtain πi = ρ0qi from (189) when evaluated on the constraint D2 = 0 and only a single term
involving the Tmn contributes to the momentum equation of motion (190) when evaluated on the
constraint D2 = 0, i.e.
πi = ρ0ηin
(Pρ0⊥
)n
r
(qm ηrsTms
)= Au
i∂
∂au
∆−1ρ0
J
(Ah`
J
∂
∂ah
(qm Ak
m
J
∂q`
∂ak
))= Au
i∂
∂au
∆−1ρ0
J
(Ah`
J
∂
∂ah
∂
∂ak
(qm Ak
m
Jq`
))= Au
i∂
∂au
∆−1ρ0
J
(1J
∂
∂ah Ah`A
km∂
∂ak
(qm q`
J
)) . (212)
Thus
qi = ηi` Au`
ρ0
∂
∂au
∆−1ρ0
J
(1J
∂
∂ah Ahj A
fk
∂
∂a f
(q j qk
J
)) = − Γ ijk(q
j, qk) , (213)
where we have swapped indices and defined Γ ijk(q
j, qk), the normal force operator for geodesic
flow, analogous to that of (62). Like Γi, jk of (63), Γ ijk possesses symmetry: given arbitrary vector
fields V and W
Γ ijk(V
j,Wk) := −ηi` Au`
ρ0
∂
∂au
∆−1ρ0
J
(1J
∂
∂ah Ahj A
fk
∂
∂a f
(V j Wk
J
)) = Γ ijk(V
k,W j) . (214)
Equation (213) defines geodesic flow on the group of volume preserving diffeomorphisms,
as was the case in Section II C 1, it does so in terms of the original coordinates, i.e., without
42
specifically transforming to normal coordinates on the constraint surfaces which here are infintie-
dimensional.
Now we are in position to close the circle by writing (213) in Eulerian form. We will do this for
the ideal fluid, but MHD follows similarly. The term qi becomes as usual the advective derivative
∂v/∂t + v · ∇v and the Γijk term gives Pρ⊥ (v · ∇v). Thus (213) is precisely the Lagrangian form of
(210), written as follows:
∂v∂t
= −Pρ (v · ∇v) = −Pρ(∇ · (v ⊗ v)
). (215)
Similarly, the Lagrangian version of (209) follows easily from (213). To see this we operate with
the counterpart of taking the Eulerian divergence on the first line of (212) and make use of (182),
Ahn
J
∂
∂ah
πn
ρ0=
Ahn
J
∂
∂ah
(Pρ0⊥
)n
r
(qm ηrsTms
)=
Ahn
J
∂
∂ah
(qm ηnsTms
)= δn
`
Ahn
J
∂
∂ah
(qm Ak
m
J
∂q`
∂ak
), (216)
which in Eulerian variables becomes
∇ ·
(∂v∂t
+ v · ∇v)
= ∇ · (v · ∇v) or ∇ ·∂v∂t
=∂
∂t∇ · v = 0 . (217)
In Lagrangian variables we have the trivial conservation laws
ρ0 = 0 and s0 = 0 , (218)
where the corresponding fluxes are identically zero. However, as is evident from (202) and (203)
we obtain nontrivial conservation laws for ρ and s with nonzero fluxes. Thus we see again, con-
sistent with Section III D, how Lagrangian and Eulerian conservation laws are not equivalent.
For the special case where ρ0 = J = 1 one could proceed directly from (7), write it in terms of
the Lagrangian variables, and obtain (213). However, without the constraint theory, one would not
immediately see it is Hamiltonian and in fact geodesic flow on an infinite-dimensional manifold.
E. Incompressible algebra of invariants
In closing this section, we examine the constants of motion for the constrained system. The
Poisson bracket together with the set of functionals that commute with the Hamiltonian, i.e., that
satisfy {H, Ia} = 0 for a = 1, 2, . . . , d, constitute the d-dimensional algebra of invariants, a subal-
gebra of the infinite-dimensional Poisson bracket realization on all functionals. This subalgebra
43
is a Lie algebra realization associated with a symmetry group of the dynamical system, and the
Poisson bracket with {Ia, · } yields the infinitesimal generators of the symmetries, i.e., the differ-
ential operator realization of the algebra. This was shown for compressible MHD in [71], where
the associated Lie algebra realization of the 10 parameter Galilean group on functionals was de-
scribed. This algebra is homomorphic to usual representations of the Galilean group, with the
Casimir invariants being in the center of the algebra composed of elements that have vanishing
Poisson bracket with all other elements.
A natural question to ask is what happens to this algebra when incompressibility is enforced by
our Dirac constraint procedure. Obviously the Hamiltonian is in the subalgebra and {H, · }∗ clearly
generates time translation, and this will be true for any Hamiltonian, but here we use Hamiltonian
of (201).
Inserting the momentum
P =
∫d3x ρv (219)
into (199) with the Hamiltonian (201) gives
{P,H}∗ = 0 (220)
without assuming ∇ · v = 0. To see this, we use (196) to obtain
PρδHδv
= ρv − ∇∆−1ρ ∇ · v and Pρ
δPi
δ3 j= ρ δi j , (221)
which when inserted into (199) gives
{Pi,H}∗ = −
∫d3x
[ρ
2∂|v|2
∂xi+ 3i∇ ·
(ρv − ∇∆−1
ρ ∇ · v)
+[(∇ × v) ×
(ρv − ∇∆−1
ρ ∇ · v)]
i
+(∇ · v)[(ρv − ∇∆−1
ρ ∇ · v)
i− ρ3i
] ]= 0 , (222)
as expected. The result of (222) follows upon using standard vector identities, integration by parts,
and the self-adjointness of ∆−1ρ .
The associated generator of space translations that satisfies the constraints is given by the op-
erator {P, · }∗, which can be shown directly. And, it follows that
{Pi, P j}∗ = 0 , ∀ i, j = 1, 2, 3 . (223)
Because the momentum contains no s dependence the the second line of (199) vanishes and using
PρδPi/δ3 j = ρ δi j of (221) it is clear the last line involving ∇ · v of (199) also vanishes. The result
of (223) is obtained because the first and third lines cancel.
44
Next, consider the angular momentum
L =
∫d3x ρ x × v . (224)
We will show
{Li,H}∗ = 0 . (225)
Using PρδLi/δv = δLi/δv, the fact that {Li,H} = 0 for the compressible fluid, and Pρ = I − Pρ⊥,
we obtain
{Li,H}∗ =
∫d3x
[− ∇
δLi
δρ· Pρ⊥(ρv)
+
(δLi
δv×∇ × vρ
+∇ · vρ
δLi
δv
)· Pρ⊥(ρv)
]. (226)
Next, recognizing that Pρ⊥(ρv) = ∇∆−1ρ ∇ · v and integrating by parts, we obtain
{Li,H}∗ =
∫d3x ∆−1
ρ (∇ · v)[∇2 δLi
δρ− ∇ ·
(δLi
δv×∇ × vρ
+δLi
δv∇ · vρ
) ]. (227)
Then upon insertingδLi
δρ= εi jkx j3k and
δLi
δ3 j= ρ xkεik j , (228)
and using standard vector analysis we obtain (225).
Because PρδLi/δv = δLi/δv, the first and third lines of (199) produce
{Li, L j}∗ = εi jkLk , (229)
just as they do for the compressible fluid (and MHD), while the fourth line manifestly vanishes.
Similarly, it follows that that {L, · }∗ is the generator for rotations.
To obtain the full algebra of invariants we need {Li, P j}∗. However because PρδPi/δv = δPi/δv
and PρδLi/δv = δLi/δv, it follows as for the compressible fluid that {Li, P j}∗ = εi jkPk.
Finally, consider the following measure of the position of the center of mass, the generator of
Galilean boosts,
G =
∫d3x ρ (x − vt) . (230)
Calculations akin to those above reveal
{Gi,G j}∗ = 0 , {Gi, P j}∗ = 0 , {Gi,H}∗ = Pi , {Li,G j}∗ = εi jkGk . (231)
Thus the bracket (207) with the set of ten invariants {H,P,L,G} is at once a closed subalge-
bra of Poisson bracket realization on all functionals and produces an operator realization of the
45
Galilean group [see e.g. 82] that is homomorphic to the operator algebra of {Li, · }∗, {Pi, · }∗, etc.
with operator commutation relations. This remains true for MHD with the only change being the
addition of HB to the Hamiltonian.
Thus, the Galilean symmetry properties of the ideal fluid and MHD are not affected by the
compressibility constraint. However, based on past experience with advected quantities, we do
expect a new Casimir invariant of the form
C[ρ, s] =
∫d3x C(ρ, s) . (232)
To see that {C, F}∗ = 0 for any functional F, where C(ρ, s) is an arbitrary function of its arguments,
we calculate
{F, C}∗ = −
∫d3x
1ρ
[ρ∇
∂C
∂ρ−∂C
∂s∇s
]· Pρ
δFδv
, (233)
and since ∇ × (ρ∇∂C/∂ρ − ∂C/∂s∇s) = 0 we write it as ∇p, giving for (233)
{F, C}∗ = −
∫d3x
1ρ∇p · Pρ
δFδv
. (234)
Thus, integration by parts and use of (198) imply {F, C}∗ = 0 for all functionals F. Note, without
loss of generality we can write C(ρ, s) = ρU(ρ, s), in which case p = ρ2∂U/∂ρ. Thus, it is
immaterial whether or not one retains the internal energy term∫
d3x ρU(ρ, s) in the Hamiltonian.
Now, (232) is not the most general Casimir. Because both ρ and ∇ · v are Lagrangian pointwise
Dirac constraints, we expect the following to be an Eulerian Casimir
C[ρ, s,∇ · v] =
∫d3xC(ρ, s,∇ · v) , (235)
where C is an arbitrary function of its arguments. To see that {C, F}∗ = 0 for any functional F, we
first observe thatδCδv
= −∇∂C
∂∇ · v(236)
and, as is evident from (196), that ∇ · (Pρ∇Φ) = 0 for all Φ; hence, all the δC/δv terms vanish
except the first term of the last line of (199). This term combines with the others to cancel, just as
for the calculation of C.
For constant density, entropy, and magnetic field, the bracket of (207) reduces to
{F,G}∗ = −
∫d3x∇ × vρ·
(PδFδv× P
δGδv
), (237)
whence it is easily seen that the helicity
C3·∇×3 =
∫d3x v · ∇ × v (238)
46
is a Casimir invariant because P (∇ × v) = ∇ × v. This Casimir is lost when entropy and density
are allowed to be advected, for it is no longer a Casimir invariant of (199).
Now, let us consider invariants in the Lagrangian description. Without the incompressibility
constraints, the Hamiltonian has a standard kinetic energy term and the internal energy depends
on ∂q/∂a, an infinitesimal version of the two-body interaction, if follows that just like the N-body
problem the system has Galilean symmetry, and because the Poisson bracket in the Lagrangian de-
scription (100) is canonical there are no Casimir invariants. With the incompressibility constraint,
the generators of the algebra now respect the constraints, with Dirac constraints being Casimirs
and the algebra of constraints now having a nontrivial center. Because the Casimirs are pointwise
invariants, we expect the situation to be like that for the Maxwell Vlasov equation [71], where the
following is a Casimir
C∇·B[B] =
∫d3xC(∇ · B, x) , (239)
where here C is an arbitrary function of its arguments. This suggests the following:
C[J] =
∫d3a C(J, a) . (240)
Indeed, only the first term of (184) contributes when we calculate {C,G}∗ and this term vanishes
by (181) becauseδCδqi = −
∂
∂a`
(A`
i∂C
∂J
). (241)
Similarly, it can be shown that the full Casimir is
C[D1,D2] =
∫d3a C(D1,D2, a) , (242)
a Lagrangian Casimir consistent with (235).
For MHD, the magnetic helicity,
CA·B =
∫d3x A · B , (243)
where B = ∇ × A is easily seen to be preserved and a Casimir up to the usual issues regarding
gauge conditions and boundary terms [see 64]. We know that the cross helicity
C3·B =
∫d3x v · B , (244)
is a Casimir of the compressible barotropic MHD equations, and it is easy to verify that it is also
a Casimir of (207) added to (208), that is for uniform density. However, it is not a Casimir for the
case with advected density, i.e., for the bracket of (199) added to (200).
47
V. CONCLUSIONS
In this paper we have substantially investigated constraints, particularly incompressibility for
the ideal fluid and MHD, for the three dichotomies described in Section I A: the Lagrangian vs.
Eulerian fluid descriptions, Lagrange multiplier vs. Dirac constraint methods, and Lagrangian vs.
Hamiltonian formalisms. An in depth description of the interplay between the various fluid and
MHD descriptions was given, with an emphasis on Dirac’s constraint method. Although we mainly
considered geodesic flow for simplicity, the Dirac’s Poisson bracket method can be used to find
other forces of constraint in a variety of fluid and plasma contexts.
Based on our results, many avenues for future research are presented. We mention a few. Since
the Hamiltonian structure of extended and relativistic MHD are now at hand [44, 58, 60, 61, 67, 70]
calculations analogous to those presented here can be done for a variety of magnetofluid models.
Another avenue for future research would be to address stability with constraints. In a previous
series of papers [45, 46, 47, 48, 49] we have investigated Hamiltonian based stability, generaliza-
tions of the MHD energy principle or the ideal fluid Rayleigh criterion, within the Lagrangian,
energy-Casimir, and dynamically accessible frameworks. Because Dirac’s method adds Casimirs,
the Dirac constraints, one gets a richer set of equilibria from the energy-Casimir variational princi-
ple and these can be tested for Lyapunov stability. Similarly, the method of dynamical accessibility
[see 72] based on constrained variations induced by the Poisson operator will enlarge the set of
stable equilibria.
Lastly, we mention that there is considerable geometric structure behind our calculations. These
can be restated in geometric/Lie group language [see e.g. 54]. Also, Arnold’s program for obtain-
ing the Riemann curvature for geodesic flow on the group of volume preserving diffeomorphisms
can be explored beginning from our results of Section IV. We did not feel this special issue would
be the appropriate place to explore these ideas.
ACKNOWLEDGMENT
PJM was supported by U.S. Dept. of Energy under contract #DE-FG02-04ER-54742. He would
also like to acknowledge support from the Humboldt Foundation and the hospitality of the Numer-
ical Plasma Physics Division of the IPP, Max Planck, Garching. FP would like to acknowledge the
48
hospitality of the Institute for Fusion Studies of the University of Texas at Austin.
[44] Abdelhamid, H. M., Kawazura, Y. & Yoshida, Z. 2015 Hamiltonian formalism of extended magneto-
hydrodynamics. J. Phys. A 48 (23), 235502.
[45] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2010 MHD equilibrium variational principles with
symmetry. Plasma Phys. and Controll. Fusion P52, 055001.
[46] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2012 Hamiltonian magnetohydrodynamics: Symmetric
formulation, Casimir invariants, and equilibrium variational principles. Phys. Plasmas 19, 052102.
[47] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2013 Hamiltonian magnetohydrodynamics: La-
grangian, Eulerian, and dynamically accessible stability - Theory. Phys. Plasmas 20, 092104.
[48] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2015 Erratum: Hamiltonian magnetohydrodynamics:
Lagrangian, Eulerian, and dynamically accessible stability - Theory. Phys. Plasmas 22, 039903.
[49] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2016 Hamiltonian magnetohydrodynamics: La-
grangian, Eulerian, and dynamically accessible stability - examples with translation symmetry. Phys.
Plasmas 23, 102112.
[50] Arnold, V. I. 1966 Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses
applications a l’hydrodynamique des fluides parfaits. Ann. l’Inst. Fourier XVI, 319–361.
[51] Arnold, V. I. 1978 Mathematical Methods of Classical Mechanics. Springer-Verlag.
[52] Arnold, V. I. & Khesin, B. A. 1998 Topological Methods in Hydrodynamics. Springer-Verlag.
[53] Arnold, V. I., Kozlov, V. V. & Neishtadt, A. I. 1980 Dynamical Systems III. Springer Verlag, New
York.
[54] Bloch, A. M. 2002 Nonholonomic Mechanics and Control. Springer-Verlag.
[55] Chandre, C., de Guillebon, L., Back, A., Tassi, E. & Morrison, P. J. 2013 On the use of projectors
for Hamiltonian systems and their relationship with Dirac brackets. J. Phys. A 46, 125203.
[56] Chandre, C., Morrison, P. J. & Tassi, E. 2012 On the Hamiltonian formulation of incompressible
ideal fluids and magnetohydrodynamics via Dirac’s theory of constraints. Phys. Lett. A 376, 737–743.
[57] Chandre, C., Morrison, P. J. & Tassi, E. 2014 Hamiltonian formulation of the modified Hasegawa-
Mima equation. Phys. Lett A 378, 956–959.
[58] Charidakos, I. Keramidas, Lingam, M., Morrison, P. J., White, R. L. & Wurm, A. 2014 Action
principles for extended MHD models. Phys. Plasmas 21, 092118.
49
[59] Corben, H. C. & Stehle, P. 1960 Classical Mechanics, 2nd edn. Wiley.
[60] D’Avignon, E., Morrison, P. J. & Pegoraro, F. 2015 Action principle for relativistic magnetohydro-
dynamics. Phys. Rev. D 91, 084050.
[61] D’Avignon, E. C., Morrison, P. J. & Lingam, M. 2016 Derivation of the Hall and extended magneto-
hydrodynamics brackets. Phys. Plasmas 23, 062101.
[62] Dirac, P. A. M. 1950 Generalized Hamiltonian dynamics. Canadian J. Math. 2, 129–148.
[63] Evans, L. C. 2010 Partial Differential Equations. American Mathematical Society, Providence, RI.
[64] Finn, J. M. & Antonsen, T. M. 1985 Magnetic helicity: What is it and what is it good for? Comments
Plasma Phys. Control. Fusion 9, 111–126.
[65] Flierl, G. R. & Morrison, P. J. 2011 Hamiltonian-Dirac simulated annealing: Application to the
calculation of vortex states. Physica D 240, 212–232.
[66] Hanson, A., Regge, T. & Teitleboim, C. 1976 Constrained Hamiltonian Systems. Accademia
Nazionale dei Lincei, Rome.
[67] Kaltsas, D. A., Throumoulopoulos, G. N. & Morrison, P. J. 2020 Energy-Casimir, dynamically
accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria. Phys. Plasmas 27,
012104.
[68] Kampen, N. G. Van& Felderhoff, B. U. 1967 Theoretical Methods in Plasma Physics. North-Holland,
Amsterdam.
[69] Lagrange, J. L. 1788 Mecanique Analytique. Ve Courcier, Paris.
[70] Lingam, M., Miloshevich, G. & Morrison, P. J. 2016 Concomitant Hamiltonian and topological struc-
tures of extended magnetohydrodynamics. Phys. Lett. A 380, 2400–2406.
[71] Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. AIP Conf. Proc. 88, 13–46.
[72] Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467–521.
[73] Morrison, P. J. 2009 On Hamiltonian and action principle formulations of plasma dynamics. AIP
Conf. Proc. 1188, 329–344.
[74] Morrison, P. J. & Greene, J. M. 1980 Noncanonical Hamiltonian density formulation of hydrody-
namics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45, 790–793.
[75] Morrison, P. J., Lebovitz, N. R. & Biello, J. 2009 The Hamiltonian description of incompressible
fluid ellipsoids. Ann. Phys. 324, 1747–1762.
[76] Newcomb, WA 1962 Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nuclear Fu-
sion Supp. 2, 451–463.
50
[77] Nguyen, S. & Turski, L. A. 1999 Canonical description of incompressible fluid: Dirac brackets ap-
proach. Physica A 272, 48–55.
[78] Nguyen, S. & Turski, L. A. 2001 Examples of the Dirac approach to dynamics of systems with
constraints. Physica A 290, 431–44.
[79] Orszag, S. A., Israeli, M. & Deville, M. O. 1986 Boundary conditions for incompressible flows. J.
Sci. Comp. 1, 75–111.
[80] Serrin, J. 1959 Mathematical principles of classical fluid mechanics. In Handbuch der Physik VIII
pt. 1 (ed. S. Flugge), pp. 125–262. Springer Verlag.
[81] Sommerfeld, A. 1964 Mechanics of Deformable Bodies. Academic Press.
[82] Sudarshan, E. C. G. & Makunda, N. 1974 Classical Dynamics : A Modern Perspective. John Wiley
& Sons, New York.
[83] Sundermeyer, K. 1982 Constrained Dynamics, Lecture Notes in Physics vol. 169. Springer Verlag,
New York.
[84] Tassi, E., Chandre, D. & Morrison, P. J. 2009 Hamiltonian derivation of the Charney-Hasegawa-
Mima equation. Phys. Plasmas 16, 082301.
[85] Van Kampen, N. G. & Felderhof, B. U. 1967 Theoretical Methods in Plasma Physics. North Holland,
Amsterdam.
[86] Whittaker, E. T. 1917 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 2nd edn.
Cambridge University Press.
[44] Abdelhamid, H. M., Kawazura, Y. & Yoshida, Z. 2015 Hamiltonian formalism of extended magneto-
hydrodynamics. J. Phys. A 48 (23), 235502.
[45] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2010 MHD equilibrium variational principles with
symmetry. Plasma Phys. and Controll. Fusion P52, 055001.
[46] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2012 Hamiltonian magnetohydrodynamics: Symmetric
formulation, Casimir invariants, and equilibrium variational principles. Phys. Plasmas 19, 052102.
[47] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2013 Hamiltonian magnetohydrodynamics: La-
grangian, Eulerian, and dynamically accessible stability - Theory. Phys. Plasmas 20, 092104.
[48] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2015 Erratum: Hamiltonian magnetohydrodynamics:
Lagrangian, Eulerian, and dynamically accessible stability - Theory. Phys. Plasmas 22, 039903.
[49] Andreussi, T., Morrison, P. J. & Pegoraro, F. 2016 Hamiltonian magnetohydrodynamics: La-
grangian, Eulerian, and dynamically accessible stability - examples with translation symmetry. Phys.
51
Plasmas 23, 102112.
[50] Arnold, V. I. 1966 Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses
applications a l’hydrodynamique des fluides parfaits. Ann. l’Inst. Fourier XVI, 319–361.
[51] Arnold, V. I. 1978 Mathematical Methods of Classical Mechanics. Springer-Verlag.
[52] Arnold, V. I. & Khesin, B. A. 1998 Topological Methods in Hydrodynamics. Springer-Verlag.
[53] Arnold, V. I., Kozlov, V. V. & Neishtadt, A. I. 1980 Dynamical Systems III. Springer Verlag, New
York.
[54] Bloch, A. M. 2002 Nonholonomic Mechanics and Control. Springer-Verlag.
[55] Chandre, C., de Guillebon, L., Back, A., Tassi, E. & Morrison, P. J. 2013 On the use of projectors
for Hamiltonian systems and their relationship with Dirac brackets. J. Phys. A 46, 125203.
[56] Chandre, C., Morrison, P. J. & Tassi, E. 2012 On the Hamiltonian formulation of incompressible
ideal fluids and magnetohydrodynamics via Dirac’s theory of constraints. Phys. Lett. A 376, 737–743.
[57] Chandre, C., Morrison, P. J. & Tassi, E. 2014 Hamiltonian formulation of the modified Hasegawa-
Mima equation. Phys. Lett A 378, 956–959.
[58] Charidakos, I. Keramidas, Lingam, M., Morrison, P. J., White, R. L. & Wurm, A. 2014 Action
principles for extended MHD models. Phys. Plasmas 21, 092118.
[59] Corben, H. C. & Stehle, P. 1960 Classical Mechanics, 2nd edn. Wiley.
[60] D’Avignon, E., Morrison, P. J. & Pegoraro, F. 2015 Action principle for relativistic magnetohydro-
dynamics. Phys. Rev. D 91, 084050.
[61] D’Avignon, E. C., Morrison, P. J. & Lingam, M. 2016 Derivation of the Hall and extended magneto-
hydrodynamics brackets. Phys. Plasmas 23, 062101.
[62] Dirac, P. A. M. 1950 Generalized Hamiltonian dynamics. Canadian J. Math. 2, 129–148.
[63] Evans, L. C. 2010 Partial Differential Equations. American Mathematical Society, Providence, RI.
[64] Finn, J. M. & Antonsen, T. M. 1985 Magnetic helicity: What is it and what is it good for? Comments
Plasma Phys. Control. Fusion 9, 111–126.
[65] Flierl, G. R. & Morrison, P. J. 2011 Hamiltonian-Dirac simulated annealing: Application to the
calculation of vortex states. Physica D 240, 212–232.
[66] Hanson, A., Regge, T. & Teitleboim, C. 1976 Constrained Hamiltonian Systems. Accademia
Nazionale dei Lincei, Rome.
[67] Kaltsas, D. A., Throumoulopoulos, G. N. & Morrison, P. J. 2020 Energy-Casimir, dynamically
accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria. Phys. Plasmas 27,
52
012104.
[68] Kampen, N. G. Van& Felderhoff, B. U. 1967 Theoretical Methods in Plasma Physics. North-Holland,
Amsterdam.
[69] Lagrange, J. L. 1788 Mecanique Analytique. Ve Courcier, Paris.
[70] Lingam, M., Miloshevich, G. & Morrison, P. J. 2016 Concomitant Hamiltonian and topological struc-
tures of extended magnetohydrodynamics. Phys. Lett. A 380, 2400–2406.
[71] Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. AIP Conf. Proc. 88, 13–46.
[72] Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467–521.
[73] Morrison, P. J. 2009 On Hamiltonian and action principle formulations of plasma dynamics. AIP
Conf. Proc. 1188, 329–344.
[74] Morrison, P. J. & Greene, J. M. 1980 Noncanonical Hamiltonian density formulation of hydrody-
namics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45, 790–793.
[75] Morrison, P. J., Lebovitz, N. R. & Biello, J. 2009 The Hamiltonian description of incompressible
fluid ellipsoids. Ann. Phys. 324, 1747–1762.
[76] Newcomb, WA 1962 Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nuclear Fu-
sion Supp. 2, 451–463.
[77] Nguyen, S. & Turski, L. A. 1999 Canonical description of incompressible fluid: Dirac brackets ap-
proach. Physica A 272, 48–55.
[78] Nguyen, S. & Turski, L. A. 2001 Examples of the Dirac approach to dynamics of systems with
constraints. Physica A 290, 431–44.
[79] Orszag, S. A., Israeli, M. & Deville, M. O. 1986 Boundary conditions for incompressible flows. J.
Sci. Comp. 1, 75–111.
[80] Serrin, J. 1959 Mathematical principles of classical fluid mechanics. In Handbuch der Physik VIII
pt. 1 (ed. S. Flugge), pp. 125–262. Springer Verlag.
[81] Sommerfeld, A. 1964 Mechanics of Deformable Bodies. Academic Press.
[82] Sudarshan, E. C. G. & Makunda, N. 1974 Classical Dynamics : A Modern Perspective. John Wiley
& Sons, New York.
[83] Sundermeyer, K. 1982 Constrained Dynamics, Lecture Notes in Physics vol. 169. Springer Verlag,
New York.
[84] Tassi, E., Chandre, D. & Morrison, P. J. 2009 Hamiltonian derivation of the Charney-Hasegawa-
Mima equation. Phys. Plasmas 16, 082301.
53
[85] Van Kampen, N. G. & Felderhof, B. U. 1967 Theoretical Methods in Plasma Physics. North Holland,
Amsterdam.
[86] Whittaker, E. T. 1917 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 2nd edn.
Cambridge University Press.
54